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  </channel><item rdf:about="https://arxiv.org/abs/2604.09723">
    <title>[2604.09723] Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields</title>
    <dc:date>2026-05-24T17:27:04+00:00</dc:date>
    <link>https://arxiv.org/abs/2604.09723</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Raz, Shalyt, Leibtag, Kalisch, Weinbaum, Hadad, and Kaminer recently showed that formulas for π can be organized by canonical polynomial recurrences and partially unified by a rank-2 Conservative Matrix Field (CMF). We prove that each order-3 recurrence explicitly printed in the public Appendix~B.6 of their paper is a shifted summation lift of an explicit order-2 kernel, and identify all three kernels: the two π-kernels are explicit rescalings of the sporadic Apéry-like sequences A036917 and A002895 (Domb numbers, case~(α)), while the Catalan kernel is a hypergeometric twist of the Gauss-square coefficient sequence at (a,b,c)=(12,1,32). We place these kernels in a unified Sym2 framework: the first π-kernel and the Catalan kernel come directly from Gauss-square coefficient sequences, while the Domb kernel is recovered by recasting the classical degree-3 Belyi pullback ϕ(x)=108x2/(1−4x)3 and the associated algebraic twist in CMF language. We write an explicit square-gauge matrix for the Gauss CMF, formulate the standard pullback--twist transport in CMF terms, and show that for rank-2 objects it is compatible with Sym2. We further prove an inverse classification: for a fixed Sym2-type Riemann scheme, the one-parameter family of Fuchsian operators contains a unique Sym2(Gauss) point, cut out by the closed-form condition λ0=2γ1γ2(1−2α) on the accessory parameter. Finally, a Belyi-pullback scan over 5040 configurations produces 11 additional integer sequences of the form [xn]λn2F1(a,b;c;ϕ(x))2; we prove their integrality and place them in the same Sym2-pullback framework.
]]></description>
<dc:subject>number-theory classification continued-fractions approximation can't-wait-to-understand-this rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e52df2334e5f/</dc:identifier>
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<item rdf:about="https://hal.science/hal-05580502v1">
    <title>Inversion of the Brillouin Function by Dynamic Geometry and Novel Continued-Fraction and Quadratic Methods - Archive ouverte HAL</title>
    <dc:date>2026-05-24T12:34:55+00:00</dc:date>
    <link>https://hal.science/hal-05580502v1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Accurate evaluation and inversion of the Brillouin function are essential for describing magnetisation in paramagnetic systems. This paper introduces three complementary computational approaches: a dynamic geometry method, a powered continued-fraction method, and a quadratic-equation approximation. The Brillouin function is first transformed into a polynomial in the variable e^(x/J), where x denotes the ratio of Zeeman energy to thermal energy and J the total angular-momentum quantum number. The polynomial is represented geometrically through line-segment relations and solved by sliding a construction point to vary the exponential variable until the geometric constraints are satisfied. Accuracy in this method is governed by practical limitations such as line thickness, finite point dimensions, compassbased measurement errors, and the adopted precision of Euler's number. Rewriting the polynomial in terms of e^(-x/J) generates an infinite recursive powered continued fraction. Since the variable remains below unity at each stage, successive powering rapidly contracts its magnitude, rendering higher-order terms negligible after a finite number of iterations. In the quadratic-approximation method, the polynomial is recast as a quadratic in a small correction variable, permitting higher-order terms to be neglected. Solved examples demonstrate that three to four iterations typically achieve precision up to ten decimal places.

]]></description>
<dc:subject>numerical-methods continued-fractions approximation algorithms rather-interesting representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:726d350e5f0b/</dc:identifier>
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<item rdf:about="https://inria.hal.science/hal-05593313v1">
    <title>Computing hard-to-round cases of sin, cos, tan in double precision - Inria - Institut national de recherche en sciences et technologies du numérique</title>
    <dc:date>2026-05-24T12:22:36+00:00</dc:date>
    <link>https://inria.hal.science/hal-05593313v1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper describes an exhaustive search algorithm to find the hardest-to-round cases of trigonometric functions (sin, cos, tan) in double precision (binary64). This algorithm reuses a clever reduction from the literature, but instead of using a sublinear search, it uses a brute force linear search. This algorithm was implemented using multi-threading and SIMD, and a full set of hard-to-round cases for the binary64 trigonometric functions was computed. As a consequence, the Table Maker's Dilemma is now fully solved for the most common univariate binary64 functions.

]]></description>
<dc:subject>numerical-methods computational-complexity algorithms rather-interesting special-cases to-write-about to-simulate approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:6249741238f1/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2302.06457">
    <title>[2302.06457] A full-stack view of probabilistic computing with p-bits: devices, architectures and algorithms</title>
    <dc:date>2026-05-24T12:06:56+00:00</dc:date>
    <link>https://arxiv.org/abs/2302.06457</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The transistor celebrated its 75th birthday in 2022. The continued scaling of the transistor defined by Moore's Law continues, albeit at a slower pace. Meanwhile, computing demands and energy consumption required by modern artificial intelligence (AI) algorithms have skyrocketed. As an alternative to scaling transistors for general-purpose computing, the integration of transistors with unconventional technologies has emerged as a promising path for domain-specific computing. In this article, we provide a full-stack review of probabilistic computing with p-bits as a representative example of the energy-efficient and domain-specific computing movement. We argue that p-bits could be used to build energy-efficient probabilistic systems, tailored for probabilistic algorithms and applications. From hardware, architecture, and algorithmic perspectives, we outline the main applications of probabilistic computers ranging from probabilistic machine learning and AI to combinatorial optimization and quantum simulation. Combining emerging nanodevices with the existing CMOS ecosystem will lead to probabilistic computers with orders of magnitude improvements in energy efficiency and probabilistic sampling, potentially unlocking previously unexplored regimes for powerful probabilistic algorithms.
]]></description>
<dc:subject>probability-theory probabilistic-computing algorithms rather-interesting machine-learning engineering-design approximation to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:77469b3f11a6/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2501.10090">
    <title>[2501.10090] Variations on a theme of Apéry</title>
    <dc:date>2026-05-24T11:05:06+00:00</dc:date>
    <link>https://arxiv.org/abs/2501.10090</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Apéry's remarkable discovery of rapidly converging continued fractions with small coefficients for ζ(2) and ζ(3) has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to show that modifications of Apéry's continued fractions can give interesting results including new rapidly convergent continued fractions for certain interesting constants.
]]></description>
<dc:subject>continued-fractions approximation number-theory elegant-mathematics rather-interesting to-write-about representation consider:symbolic-regression</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f73b97ad1bf4/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2502.12717">
    <title>[2502.12717] Learning the symmetric group: large from small</title>
    <dc:date>2026-05-22T11:20:50+00:00</dc:date>
    <link>https://arxiv.org/abs/2502.12717</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Machine learning explorations can make significant inroads into solving difficult problems in pure mathematics. One advantage of this approach is that mathematical datasets do not suffer from noise, but a challenge is the amount of data required to train these models and that this data can be computationally expensive to generate. Key challenges further comprise difficulty in a posteriori interpretation of statistical models and the implementation of deep and abstract mathematical problems.
We propose a method for scalable tasks, by which models trained on simpler versions of a task can then generalize to the full task. Specifically, we demonstrate that a transformer neural-network trained on predicting permutations from words formed by general transpositions in the symmetric group S10 can generalize to the symmetric group S25 with near 100\% accuracy. We also show that S10 generalizes to S16 with similar performance if we only use adjacent transpositions. We employ identity augmentation as a key tool to manage variable word lengths, and partitioned windows for training on adjacent transpositions. Finally we compare variations of the method used and discuss potential challenges with extending the method to other tasks.
]]></description>
<dc:subject>machine-learning mathematical-programming combinatorics neural-networks formal-languages to-understand to-write-about group-theory rather-interesting feature-construction approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:062e950d111d/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2601.10667">
    <title>[2601.10667] Stable evaluation of derivatives for barycentric and continued fraction representations of rational functions</title>
    <dc:date>2026-01-21T12:46:20+00:00</dc:date>
    <link>https://arxiv.org/abs/2601.10667</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Fast algorithms for approximation by rational functions exist for both barycentric and Thiele continued fraction (TCF) representations. We present the first numerically stable methods for derivative evaluation in the barycentric representation, including an O(n) algorithm for all derivatives. We also extend an earlier O(n) algorithm for evaluation of the TCF first derivative to higher orders. Numerical experiments confirm the robustness and efficiency of the proposed methods.
]]></description>
<dc:subject>numerical-methods approximation rather-interesting algorithms continued-fractions representation to-write-about to-simulate consider:constructive-generalizations</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f185ff1f75fd/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2510.16250">
    <title>[2510.16250] One-Bit Quantization for Random Features Models</title>
    <dc:date>2026-01-18T21:11:47+00:00</dc:date>
    <link>https://arxiv.org/abs/2510.16250</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Recent advances in neural networks have led to significant computational and memory demands, spurring interest in one-bit weight compression to enable efficient inference on resource-constrained devices. However, the theoretical underpinnings of such compression remain poorly understood. We address this gap by analyzing one-bit quantization in the Random Features model, a simplified framework that corresponds to neural networks with random representations. We prove that, asymptotically, quantizing weights of all layers except the last incurs no loss in generalization error, compared to the full precision random features model. Our findings offer theoretical insights into neural network compression. We also demonstrate empirically that one-bit quantization leads to significant inference speed ups for the Random Features models even on a laptop GPU, confirming the practical benefits of our work. Additionally, we provide an asymptotically precise characterization of the generalization error for Random Features with an arbitrary number of layers. To the best of our knowledge, our analysis yields more general results than all previous works in the related literature.
]]></description>
<dc:subject>approximation compressed-sensing compression neural-networks rather-interesting to-understand to-simulate consider:symbolic-regression consider:expression-trees</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:004d00cd5297/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2005.07678">
    <title>[2005.07678] Edit Distance in Near-Linear Time: it's a Constant Factor</title>
    <dc:date>2025-12-31T17:56:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2005.07678</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present an algorithm for approximating the edit distance between two strings of length n in time n1+ε up to a constant factor, for any ε>0. Our result completes a research direction set forth in the recent breakthrough paper [Chakraborty-Das-Goldenberg-Koucky-Saks, FOCS'18], which showed the first constant-factor approximation algorithm with a (strongly) sub-quadratic running time. The recent results of [Koucky-Saks, STOC'20] and [Brakensiek-Rubinstein, STOC'20] have shown near-linear time algorithms that obtain an additive approximation, near-linear in n (equivalently, constant-factor approximation when the edit distance value is close to n). In contrast, our algorithm obtains a constant-factor approximation in near-linear time for any input strings.
In contrast to prior algorithms, which are mostly recursing over smaller substrings, our algorithm gradually smoothes out the local contribution to the edit distance over progressively larger substrings. To accomplish this, we iteratively construct a distance oracle data structure for the metric of edit distance on all substrings of input strings, of length niε for i=0,1,…,1/ε. The distance oracle approximates the edit distance over these substrings in a certain average sense, just enough to estimate the overall edit distance.
]]></description>
<dc:subject>strings computational-complexity algorithms rather-interesting approximation to-write-about to-implement consider:Push-code</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:71df08420968/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:strings"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-implement"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:Push-code"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2308.06277">
    <title>[2308.06277] Descriptive complexity for neural networks via Boolean networks</title>
    <dc:date>2025-12-01T15:55:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2308.06277</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We investigate the expressive power of neural networks from the point of view of descriptive complexity. We study neural networks that use floating-point numbers and piecewise polynomial activation functions from two perspectives: 1) the general scenario where neural networks run for an unlimited number of rounds and have unrestricted topologies, and 2) classical feedforward neural networks that have the topology of layered acyclic graphs and run for only a constant number of rounds. We characterize these neural networks via Boolean networks formalized via a recursive rule-based logic. In particular, we show that the sizes of the neural networks and the corresponding Boolean rule formulae are polynomially related. In fact, in the direction from Boolean rules to neural networks, the blow-up is only linear. Our translations result in a time delay, which is the number of rounds that it takes to simulate a single computation step. In the translation from neural networks to Boolean rules, the time delay of the resulting formula is polylogarithmic in the size of the neural network. In the converse translation, the time delay of the neural network is linear in the formula size. Ultimately, we obtain translations between neural networks, Boolean networks, the diamond-free fragment of modal substitution calculus, and a class of recursive Boolean circuits. Our translations offer a method, for almost any activation function F, of translating any neural network in our setting into an equivalent neural network that uses F at each node. This even includes linear activation functions, which is possible due to using floats rather than actual reals!
]]></description>
<dc:subject>boolean-networks neural-networks approximation rather-interesting nonlinear-dynamics complexology to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d77e837ab688/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:boolean-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:complexology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2509.01105">
    <title>[2509.01105] Distance between cubics and rationals</title>
    <dc:date>2025-11-01T20:35:25+00:00</dc:date>
    <link>https://arxiv.org/abs/2509.01105</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We investigate the following problem: what is the smallest possible distance between a cubic irrational ξ and a rational number p/q in terms of the height H(ξ) and q? More precisely, we consider the set D3,1 of all pairs (u,v) of positive real numbers such that |ξ−p/q|>cH−u(ξ)q−v for all cubic irrationals ξ and rationals p/q. First, we transform this problem into one about the root separation of cubic polynomials. Second, we establish a conditional result: under the Hall conjecture (a special case of the famous abc-conjecture), every pair (u,v)=(3+2ϵ,5/2+ϵ) belongs to D3,1. Third, we show that every pair (u,v)∈D3,1 must satisfy u≥10−3v. Finally, we discuss an analogue of the set D3,1 in function fields where we are able to give a complete description.
]]></description>
<dc:subject>number-theory approximation numerical-methods rather-interesting continued-fractions to-write-about to-visualize consider:polynomial-approximations</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f1bcb2412e97/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:polynomial-approximations"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2508.19743">
    <title>[2508.19743] Superoptimal continued fractions</title>
    <dc:date>2025-09-06T15:14:44+00:00</dc:date>
    <link>https://arxiv.org/abs/2508.19743</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Motivated by the optimal continued fractions studied independently by Selenius and Bosma, we define and introduce algorithms producing superoptimal continued fraction expansions of irrationals. These expansions simultaneously provide arbitrarily good rational approximations and converge arbitrarily quickly.
]]></description>
<dc:subject>nonlinear-dynamics continued-fractions approximation representation to-write-about to-simulate consider:functions-on-maps</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ba3041fef470/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:functions-on-maps"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2209.15542">
    <title>[2209.15542] The worst approximable rational numbers</title>
    <dc:date>2025-08-11T13:51:03+00:00</dc:date>
    <link>https://arxiv.org/abs/2209.15542</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We classify and enumerate all rational numbers with approximation constant at least 13 using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.
]]></description>
<dc:subject>number-theory approximation rational-numbers rather-interesting looking-to-see computational-complexity to-write-about to-simulate representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:73b9846f5700/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rational-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://link.springer.com/article/10.1007/s11047-025-10028-7">
    <title>Genetic programming policies for bin packing in the framework of deterministic Markov decision process | Natural Computing</title>
    <dc:date>2025-07-24T11:17:41+00:00</dc:date>
    <link>https://link.springer.com/article/10.1007/s11047-025-10028-7</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The Bin Packing Problem (BPP) is a well-known NP-hard problem with numerous real-world applications. This study focuses on minimizing waste and maximum lateness in a one-dimensional version of the BPP, which is particularly relevant in industrial contexts. The goal is to develop a constructive heuristic algorithm that can be adapted to various situations. We model the BPP as a Deterministic Markov Decision Process with discrete state and action spaces, where policies are represented by arithmetic expressions involving state variables. This approach allows for a clearer explanation of the decision process, in contrast to other methods like neural networks. To evolve these policies, we use Genetic Programming (GP). Trained on a set of BPP instances, the resulting policies are effective for solving new, unseen instances. In the experimental study, we explore different GP settings, including varying sets of symbols. The results reveal valuable insights about the importance of state variables, indicating that a smaller selection of them may yield the best results. The evolved policies are compared with an exact method from the literature, achieving similar outcomes but with significantly less computational time.

]]></description>
<dc:subject>operations-research packing genetic-programming rather-interesting deterministic-approximations approximation machine-learning to-write-about consider:heuristic-move</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:19868cd2a237/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:operations-research"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:deterministic-approximations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:heuristic-move"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.pnas.org/doi/10.1073/pnas.0710743106">
    <title>Efficient computation of optimal actions | PNAS</title>
    <dc:date>2025-05-27T22:44:00+00:00</dc:date>
    <link>https://www.pnas.org/doi/10.1073/pnas.0710743106</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Optimal choice of actions is a fundamental problem relevant to fields as diverse as neuroscience, psychology, economics, computer science, and control engineering. Despite this broad relevance the abstract setting is similar: we have an agent choosing actions over time, an uncertain dynamical system whose state is affected by those actions, and a performance criterion that the agent seeks to optimize. Solving problems of this kind remains hard, in part, because of overly generic formulations. Here, we propose a more structured formulation that greatly simplifies the construction of optimal control laws in both discrete and continuous domains. An exhaustive search over actions is avoided and the problem becomes linear. This yields algorithms that outperform Dynamic Programming and Reinforcement Learning, and thereby solve traditional problems more efficiently. Our framework also enables computations that were not possible before: composing optimal control laws by mixing primitives, applying deterministic methods to stochastic systems, quantifying the benefits of error tolerance, and inferring goals from behavioral data via convex optimization. Development of a general class of easily solvable problems tends to accelerate progress—as linear systems theory has done, for example. Our framework may have similar impact in fields where optimal choice of actions is relevant.]]></description>
<dc:subject>machine-learning approximation algorithms rather-interesting computational-complexity to-understand mathematical-programming metaheuristics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d4e94025ab87/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:metaheuristics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2407.11533">
    <title>[2407.11533] Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem</title>
    <dc:date>2024-12-21T17:43:43+00:00</dc:date>
    <link>https://arxiv.org/abs/2407.11533</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain L∞-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose L∞ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.
]]></description>
<dc:subject>low-discrepancy-samples algorithms numerical-methods approximation rather-interesting to-write-about to-simulate combinatorics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1f07890f97e1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-samples"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2408.06475">
    <title>[2408.06475] Quasi-Monte Carlo Beyond Hardy-Krause</title>
    <dc:date>2024-12-21T17:39:52+00:00</dc:date>
    <link>https://arxiv.org/abs/2408.06475</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The classical approaches to numerically integrating a function f are Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods. MC methods use random samples to evaluate f and have error O(σ(f)/n‾√), where σ(f) is the standard deviation of f. QMC methods are based on evaluating f at explicit point sets with low discrepancy, and as given by the classical Koksma-Hlawka inequality, they have error O˜(σ𝖧𝖪(f)/n), where σ𝖧𝖪(f) is the variation of f in the sense of Hardy and Krause. These two methods have distinctive advantages and shortcomings, and a fundamental question is to find a method that combines the advantages of both.
In this work, we give a simple randomized algorithm that produces QMC point sets with the following desirable features: (1) It achieves substantially better error than given by the classical Koksma-Hlawka inequality. In particular, it has error O˜(σ𝖲𝖮(f)/n), where σ𝖲𝖮(f) is a new measure of variation that we introduce, which is substantially smaller than the Hardy-Krause variation. (2) The algorithm only requires random samples from the underlying distribution, which makes it as flexible as MC. (3) It automatically achieves the best of both MC and QMC (and the above improvement over Hardy-Krause variation) in an optimal way. (4) The algorithm is extremely efficient, with an amortized O˜(1) runtime per sample.
Our method is based on the classical transference principle in geometric discrepancy, combined with recent algorithmic innovations in combinatorial discrepancy that besides producing low-discrepancy colorings, also guarantee certain subgaussian properties. This allows us to bypass several limitations of previous works in bridging the gap between MC and QMC methods and go beyond the Hardy-Krause variation.
]]></description>
<dc:subject>low-discrepancy-numbers numerical-methods integration algorithms rather-interesting number-theory probability-theory to-understand to-simulate approximation sampling</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f356a8ba7caa/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:integration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2404.05259">
    <title>[2404.05259] Cellular automata, many-valued logic, and deep neural networks</title>
    <dc:date>2024-10-30T12:32:39+00:00</dc:date>
    <link>https://arxiv.org/abs/2404.05259</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We develop a theory characterizing the fundamental capability of deep neural networks to learn, from evolution traces, the logical rules governing the behavior of cellular automata (CA). This is accomplished by first establishing a novel connection between CA and Lukasiewicz propositional logic. While binary CA have been known for decades to essentially perform operations in Boolean logic, no such relationship exists for general CA. We demonstrate that many-valued (MV) logic, specifically Lukasiewicz propositional logic, constitutes a suitable language for characterizing general CA as logical machines. This is done by interpolating CA transition functions to continuous piecewise linear functions, which, by virtue of the McNaughton theorem, yield formulae in MV logic characterizing the CA. Recognizing that deep rectified linear unit (ReLU) networks realize continuous piecewise linear functions, it follows that these formulae are naturally extracted from CA evolution traces by deep ReLU networks. A corresponding algorithm together with a software implementation is provided. Finally, we show that the dynamical behavior of CA can be realized by recurrent neural networks.
]]></description>
<dc:subject>neural-networks cellular-automata learning-by-watching recurrent-networks nonlinear-dynamics approximation machine-learning to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:39d241f4a9a0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:cellular-automata"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:learning-by-watching"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:recurrent-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2306.16998">
    <title>[2306.16998] Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms</title>
    <dc:date>2024-10-20T16:10:17+00:00</dc:date>
    <link>https://arxiv.org/abs/2306.16998</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The L∞ star discrepancy is a measure for the regularity of a finite set of points taken from [0,1)d. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other applications. Unfortunately, computing the L∞ star discrepancy of a given point set is known to be a hard problem, with the best exact algorithms falling short for even moderate dimensions around 8. However, despite the difficulty of finding the global maximum that defines the L∞ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches.
In this work we compare 8 popular numerical black-box optimization algorithms on the L∞ star discrepancy computation problem, using a wide set of instances in dimensions 2 to 15. We show that all used optimizers perform very badly on a large majority of the instances and that in many cases random search outperforms even the more sophisticated solvers. We suspect that state-of-the-art numerical black-box optimization techniques fail to capture the global structure of the problem, an important shortcoming that may guide their future development.
We also provide a parallel implementation of the best-known algorithm to compute the discrepancy.
]]></description>
<dc:subject>discrepancy approximation numerical-methods randomness quasirandom-numbers monte-carlo-methods sampling visualization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ee82b8852476/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:discrepancy"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:randomness"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:quasirandom-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:monte-carlo-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2203.14614">
    <title>[2203.14614] Sublinear-Time Probabilistic Cellular Automata</title>
    <dc:date>2024-10-13T22:09:17+00:00</dc:date>
    <link>https://arxiv.org/abs/2203.14614</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one- and two-sided error versions of the model (in the same spirit as the classes 𝖱𝖯 and 𝖡𝖯𝖯) and establish a separation between the classes of languages they can recognize all the way up to o(n‾√) time. As a consequence, we have a Ω(n‾√) lower bound for derandomizing constant-time two-sided error PACAs (using deterministic ACAs). We also prove that derandomization of T(n)-time PACAs (to polynomial-time deterministic cellular automata) for various regimes of T(n)=ω(logn) implies non-trivial derandomization results for the class 𝖱𝖯 (e.g., 𝖯=𝖱𝖯). The main contribution is an almost full characterization of the constant-time PACA classes: For one-sided error, the class equals that of the deterministic model; that is, constant-time one-sided error PACAs can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we identify a natural class we call the linearly testable languages (𝖫𝖫𝖳) and prove that the languages decidable by constant-time two-sided error PACAs are "sandwiched" in-between the closure of 𝖫𝖫𝖳 under union and intersection and the class of locally threshold testable languages (𝖫𝖳𝖳).
]]></description>
<dc:subject>distributed-processing asynchronous-behavior collective-behavior cellular-automata algorithms rather-interesting approximation to-write-about to-simulate consider:ACA-as-a-language</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:3966cfb54e9d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:distributed-processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:asynchronous-behavior"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:collective-behavior"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:cellular-automata"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ACA-as-a-language"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2410.04264">
    <title>[2410.04264] Visualising Feature Learning in Deep Neural Networks by Diagonalizing the Forward Feature Map</title>
    <dc:date>2024-10-08T19:58:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2410.04264</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Deep neural networks (DNNs) exhibit a remarkable ability to automatically learn data representations, finding appropriate features without human input. Here we present a method for analysing feature learning by decomposing DNNs into 1) a forward feature-map Φ that maps the input dataspace to the post-activations of the penultimate layer, and 2) a final linear layer that classifies the data. We diagonalize Φ with respect to the gradient descent operator and track feature learning by measuring how the eigenfunctions and eigenvalues of Φ change during training. Across many popular architectures and classification datasets, we find that DNNs converge, after just a few epochs, to a minimal feature (MF) regime dominated by a number of eigenfunctions equal to the number of classes. This behaviour resembles the neural collapse phenomenon studied at longer training times. For other DNN-data combinations, such as a fully connected network on CIFAR10, we find an extended feature (EF) regime where significantly more features are used. Optimal generalisation performance upon hyperparameter tuning typically coincides with the MF regime, but we also find examples of poor performance within the MF regime. Finally, we recast the phenomenon of neural collapse into a kernel picture which can be extended to broader tasks such as regression.
]]></description>
<dc:subject>deep-learning neural-networks approximation rather-interesting to-understand hey-I-know-this-guy</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4689073768de/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:deep-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:hey-I-know-this-guy"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2210.14132">
    <title>[2210.14132] p-adic Cellular Neural Networks: Applications to Image Processing</title>
    <dc:date>2024-09-28T15:43:03+00:00</dc:date>
    <link>https://arxiv.org/abs/2210.14132</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The p-adic cellular neural networks (CNNs) are mathematical generalizations of the neural networks introduced by Chua and Yang in the 80s. In this work we present two new types of CNNs that can perform computations with real data, and whose dynamics can be understood almost completely. The first type of networks are edge detectors for grayscale images. The stationary states of these networks are organized hierarchically in a lattice structure. The dynamics of any of these networks consists of transitions toward some minimal state in the lattice. The second type is a new class of reaction-diffusion networks. We investigate the stability of these networks and show that they can be used as filters to reduce noise, preserving the edges, in grayscale images polluted with additive Gaussian noise. The networks introduced here were found experimentally. They are abstract evolution equations on spaces of real-valued functions defined in the p-adic unit ball for some prime number p. In practical applications the prime p is determined by the size of image, and thus, only small primes are used. We provide several numerical simulations showing how these networks work.
]]></description>
<dc:subject>rather-interesting p-adic-numbers cellular-automata to-understand image-processing approximation algorithms representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:32e459800586/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:p-adic-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:cellular-automata"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:image-processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2402.18014">
    <title>[2402.18014] Set-valued Star-Shaped Risk Measures</title>
    <dc:date>2024-09-23T13:18:27+00:00</dc:date>
    <link>https://arxiv.org/abs/2402.18014</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of set-valued normalized convex risk measures. The link between set-valued risk measures and set-valued star-shaped risk measures is also established.
]]></description>
<dc:subject>approximation numerical-methods statistics risk rather-interesting portfolio-theory to-understand geometry error</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2b593313b435/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:risk"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:portfolio-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:error"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2207.13944">
    <title>[2207.13944] On the Multidimensional Random Subset Sum Problem</title>
    <dc:date>2024-09-21T13:58:51+00:00</dc:date>
    <link>https://arxiv.org/abs/2207.13944</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In the Random Subset Sum Problem, given n i.i.d. random variables X1,...,Xn, we wish to approximate any point z∈[−1,1] as the sum of a suitable subset Xi1(z),...,Xis(z) of them, up to error ε. Despite its simple statement, this problem is of fundamental interest to both theoretical computer science and statistical mechanics. More recently, it gained renewed attention for its implications in the theory of Artificial Neural Networks. An obvious multidimensional generalisation of the problem is to consider n i.i.d. d-dimensional random vectors, with the objective of approximating every point z∈[−1,1]d. In 1998, G. S. Lueker showed that, in the one-dimensional setting, n=(log1ε) samples guarantee the approximation property with high this http URL this work, we prove that, in d dimensions, n=(d3log1ε⋅(log1ε+logd)) samples suffice for the approximation property to hold with high probability. As an application highlighting the potential interest of this result, we prove that a recently proposed neural network model exhibits universality: with high probability, the model can approximate any neural network within a polynomial overhead in the number of parameters.
]]></description>
<dc:subject>probability-theory number-theory approximation rather-interesting to-simulate consider:lexicase-selection consider:numerical-modeling consider:quasirandom-numbers</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:5a4d9bfcab6c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:lexicase-selection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:numerical-modeling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:quasirandom-numbers"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2111.10722">
    <title>[2111.10722] A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel</title>
    <dc:date>2024-08-02T01:19:43+00:00</dc:date>
    <link>https://arxiv.org/abs/2111.10722</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We propose a novel deterministic sampling method to approximate a target distribution ρ∗ by minimizing the kernel discrepancy, also known as the Maximum Mean Discrepancy (MMD). By employing the general \emph{energetic variational inference} framework (Wang et al., 2021), we convert the problem of minimizing MMD to solving a dynamic ODE system of the particles. We adopt the implicit Euler numerical scheme to solve the ODE systems. This leads to a proximal minimization problem in each iteration of updating the particles, which can be solved by optimization algorithms such as L-BFGS. The proposed method is named EVI-MMD. To overcome the long-existing issue of bandwidth selection of the Gaussian kernel, we propose a novel way to specify the bandwidth dynamically. Through comprehensive numerical studies, we have shown the proposed adaptive bandwidth significantly improves the EVI-MMD. We use the EVI-MMD algorithm to solve two types of sampling problems. In the first type, the target distribution is given by a fully specified density function. The second type is a "two-sample problem", where only training data are available. The EVI-MMD method is used as a generative learning model to generate new samples that follow the same distribution as the training data. With the recommended settings of the tuning parameters, we show that the proposed EVI-MMD method outperforms some existing methods for both types of problems.
]]></description>
<dc:subject>machine-learning discrepancy-theory approximation sampling really-very-interesting to-write-about consider:combinatorics consider:algorithmic-information-theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:b9864fa33c22/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:discrepancy-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:really-very-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:algorithmic-information-theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2402.17764">
    <title>[2402.17764] The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits</title>
    <dc:date>2024-03-31T00:33:08+00:00</dc:date>
    <link>https://arxiv.org/abs/2402.17764</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Recent research, such as BitNet, is paving the way for a new era of 1-bit Large Language Models (LLMs). In this work, we introduce a 1-bit LLM variant, namely BitNet b1.58, in which every single parameter (or weight) of the LLM is ternary {-1, 0, 1}. It matches the full-precision (i.e., FP16 or BF16) Transformer LLM with the same model size and training tokens in terms of both perplexity and end-task performance, while being significantly more cost-effective in terms of latency, memory, throughput, and energy consumption. More profoundly, the 1.58-bit LLM defines a new scaling law and recipe for training new generations of LLMs that are both high-performance and cost-effective. Furthermore, it enables a new computation paradigm and opens the door for designing specific hardware optimized for 1-bit LLMs.
]]></description>
<dc:subject>neural-networks compression rather-interesting representation approximation huh-wouldja-lookit-that</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4707a0ccd457/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:compression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:huh-wouldja-lookit-that"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2209.01147">
    <title>[2209.01147] Algorithms for Discrepancy, Matchings, and Approximations: Fast, Simple, and Practical</title>
    <dc:date>2023-10-10T10:04:37+00:00</dc:date>
    <link>https://arxiv.org/abs/2209.01147</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study one of the key tools in data approximation and optimization: low-discrepancy colorings. Formally, given a finite set system (X,), the \emph{discrepancy} of a two-coloring χ:X→{−1,1} is defined as maxS∈|χ(S)|, where χ(S)=∑x∈Sχ(x).
We propose a randomized algorithm which, for any d>0 and (X,) with dual shatter function π∗(k)=O(kd), returns a coloring with expected discrepancy O(|X|1−1/dlog||‾‾‾‾‾‾‾‾‾‾‾‾‾‾√) (this bound is tight) in time Õ(||⋅|X|1/d+|X|2+1/d), improving upon the previous-best time of O(||⋅|X|3) by at least a factor of |X|2−1/d when ||≥|X|. This setup includes many geometric classes, families of bounded dual VC-dimension, and others. As an immediate consequence, we obtain an improved algorithm to construct ε-approximations of sub-quadratic size.
Our method uses primal-dual reweighing with an improved analysis of randomly updated weights and exploits the structural properties of the set system via matchings with low crossing number -- a fundamental structure in computational geometry. In particular, we get the same |X|2−1/d factor speed-up on the construction time of matchings with crossing number O(|X|1−1/d), which is the first improvement since the 1980s.
The proposed algorithms are very simple, which makes it possible, for the first time, to compute colorings with near-optimal discrepancies and near-optimal sized approximations for abstract and geometric set systems in dimensions higher than 2.
]]></description>
<dc:subject>low-discrepancy-samples sampling approximation algorithms rather-interesting to-understand to-visualize to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2c854195ad31/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-samples"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2209.14440">
    <title>[2209.14440] GeONet: a neural operator for learning the Wasserstein geodesic</title>
    <dc:date>2023-09-30T12:46:11+00:00</dc:date>
    <link>https://arxiv.org/abs/2209.14440</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Optimal transport (OT) offers a versatile framework to compare complex data distributions in a geometrically meaningful way. Traditional methods for computing the Wasserstein distance and geodesic between probability measures require mesh-dependent domain discretization and suffer from the curse-of-dimensionality. We present GeONet, a mesh-invariant deep neural operator network that learns the non-linear mapping from the input pair of initial and terminal distributions to the Wasserstein geodesic connecting the two endpoint distributions. In the offline training stage, GeONet learns the saddle point optimality conditions for the dynamic formulation of the OT problem in the primal and dual spaces that are characterized by a coupled PDE system. The subsequent inference stage is instantaneous and can be deployed for real-time predictions in the online learning setting. We demonstrate that GeONet achieves comparable testing accuracy to the standard OT solvers on simulation examples and the MNIST dataset with considerably reduced inference-stage computational cost by orders of magnitude.
]]></description>
<dc:subject>machine-learning probability-theory rather-interesting metrics distance numerical-methods approximation neural-networks consider:looking-to-see consider:representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2550a216a21d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:metrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:distance"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2109.10529">
    <title>[2109.10529] Numerical Continued Fraction Interpolation</title>
    <dc:date>2023-09-09T22:36:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2109.10529</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We show that highly accurate approximations can often be obtained from constructing Thiele interpolating continued fractions by a Greedy selection of the interpolation points together with an early termination condition. The obtained results are comparable with the outcome from state-of-the-art rational interpolation techniques based on the barycentric form.
]]></description>
<dc:subject>continued-fractions approximation numerical-methods rather-interesting to-understand representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:3a16dc1b8540/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2109.07838">
    <title>[2109.07838] Computing the exact sign of sums of products with floating point arithmetic</title>
    <dc:date>2023-09-09T22:35:16+00:00</dc:date>
    <link>https://arxiv.org/abs/2109.07838</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[IIn computational geometry, the construction of essential primitives like convex hulls, Voronoi diagrams and Delaunay triangulations require the evaluation of the signs of determinants, which are sums of products. The same signs are needed for the exact solution of linear programming problems and systems of linear inequalities. Computing these signs exactly with inexact floating point arithmetic is challenging, and we present yet another algorithm for this task. Our algorithm is efficient and uses only of floating point arithmetic, which is much faster than exact arithmetic. We prove that the algorithm is correct and provide efficient and tested \texttt{C++} code for it.
]]></description>
<dc:subject>numerical-methods floating-point approximation error rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e1c6d71a6a09/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:floating-point"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:error"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2308.04825">
    <title>[2308.04825] Repelled point processes with application to numerical integration</title>
    <dc:date>2023-09-09T13:01:14+00:00</dc:date>
    <link>https://arxiv.org/abs/2308.04825</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Linear statistics of point processes yield Monte Carlo estimators of integrals. While the simplest approach relies on a homogeneous Poisson point process, more regularly spread point processes, such as scrambled low-discrepancy sequences or determinantal point processes, can yield Monte Carlo estimators with fast-decaying mean square error. Following the intuition that more regular configurations result in lower integration error, we introduce the repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our main theoretical result is that applying the repulsion operator to a homogeneous Poisson point process yields an unbiased Monte Carlo estimator with lower variance than under the original point process. On the computational side, the evaluation of our estimator is only quadratic in the number of integrand evaluations and can be easily parallelized without any communication across tasks. We illustrate our variance reduction result with numerical experiments and compare it to popular Monte Carlo methods. Finally, we numerically investigate a few open questions on the repulsion operator. In particular, the experiments suggest that the variance reduction also holds when the operator is applied to other motion-invariant point processes.
]]></description>
<dc:subject>low-discrepancy-sequence numerical-methods sampling algorithms horse-races rather-interesting approximation performance-measure nonlinear-dynamics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1d714b506f44/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-sequence"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:horse-races"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:performance-measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2002.06118">
    <title>[2002.06118] Covering of high-dimensional cubes and quantization</title>
    <dc:date>2023-08-19T22:16:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2002.06118</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[As the main problem, we consider covering of a d-dimensional cube by n balls with reasonably large d (10 or more) and reasonably small n, like n=100 or n=1000. We do not require the full coverage but only 90\% or 95\% coverage. We establish that efficient covering schemes have several important properties which are not seen in small dimensions and in asymptotical considerations, for very large n. One of these properties can be termed `do not try to cover the vertices' as the vertices of the cube and their close neighbourhoods are very hard to cover and for large d there are far too many of them. We clearly demonstrate that, contrary to a common belief, placing balls at points which form a low-discrepancy sequence in the cube, makes for a very inefficient covering scheme. For a family of random coverings, we are able to provide very accurate approximations to the coverage probability. We then extend our results to the problems of coverage of a cube by smaller cubes and quantization, the latter being also referred to as facility location. Along with theoretical considerations and derivation of approximations, we discuss results of a large-scale numerical investigation.
]]></description>
<dc:subject>computational-geometry low-discrepancy-numbers approximation looking-to-see geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0df86e75f223/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2102.07833">
    <title>[2102.07833] Quasi-Monte Carlo Software</title>
    <dc:date>2023-08-19T21:39:09+00:00</dc:date>
    <link>https://arxiv.org/abs/2102.07833</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Practitioners wishing to experience the efficiency gains from using low discrepancy sequences need correct, robust, well-written software. This article, based on our MCQMC 2020 tutorial, describes some of the better quasi-Monte Carlo (QMC) software available. We highlight the key software components required by QMC to approximate multivariate integrals or expectations of functions of vector random variables. We have combined these components in QMCPy, a Python open-source library, which we hope will draw the support of the QMC community. Here we introduce QMCPy.
]]></description>
<dc:subject>numerical-methods low-discrepancy-numbers algorithms tutorial approximation rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a246e9d597c9/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tutorial"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2210.16548">
    <title>[2210.16548] The importance of being scrambled: supercharged Quasi Monte Carlo</title>
    <dc:date>2023-08-19T14:16:28+00:00</dc:date>
    <link>https://arxiv.org/abs/2210.16548</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In many financial applications Quasi Monte Carlo (QMC) based on Sobol low-discrepancy sequences (LDS) outperforms Monte Carlo showing faster and more stable convergence. However, unlike MC QMC lacks a practical error estimate. Randomized QMC (RQMC) method combines the best of two methods. Application of scrambled LDS allow to compute confidence intervals around the estimated value, providing a practical error bound. Randomization of Sobol' LDS by two methods: Owen's scrambling and digital shift are compared considering computation of Asian options and Greeks using hyperbolic local volatility model. RQMC demonstrated the superior performance over standard QMC showing increased convergence rates and providing practical error bounds.
]]></description>
<dc:subject>approximation numerical-methods low-discrepancy quasirandom-numbers rather-interesting heuristics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ded4a361ce8c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:low-discrepancy"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:quasirandom-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:heuristics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2110.14466">
    <title>[2110.14466] Random Lochs' Theorem</title>
    <dc:date>2023-02-07T13:03:55+00:00</dc:date>
    <link>https://arxiv.org/abs/2110.14466</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number x that can be determined from just knowing its first n decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs' Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin's formula for entropy.
]]></description>
<dc:subject>number-theory information-theory rather-interesting representation continued-fractions approximation heuristics to-understand to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:29fd6e3eb76f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:information-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:heuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2109.08962">
    <title>[2109.08962] On integer partitions and continued fraction type algorithms</title>
    <dc:date>2023-02-07T13:01:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2109.08962</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number n into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for p(2,n), the number of ways for partitioning n into two smaller positive integers, with multiplicity. We then do the analogue using the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm), giving us a method for producing integer partitions of a positive number n into three smaller numbers, with multiplicity. However different aspects of this generalisations remain unclear.
]]></description>
<dc:subject>number-theory enumeration continued-fractions rather-interesting to-understand representation approximation optimization computational-complexity</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2d5320455881/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:enumeration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2203.01794">
    <title>[2203.01794] Efficient Fréchet distance queries for segments</title>
    <dc:date>2023-01-01T13:09:52+00:00</dc:date>
    <link>https://arxiv.org/abs/2203.01794</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab⎯⎯⎯⎯⎯⎯ one can efficiently compute the Fréchet distance between P and ab⎯⎯⎯⎯⎯⎯. First we present a data structure of size O(nlogn) that can compute the Fréchet distance between P and a horizontal query segment ab⎯⎯⎯⎯⎯⎯ in O(logn) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab⎯⎯⎯⎯⎯⎯ together with two points s,t∈P (not necessarily vertices), and ask for the Fréchet distance between ab⎯⎯⎯⎯⎯⎯ and the curve of P in between s and t. Using O(nlog2n) storage, such queries take O(log3n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk3+ε+n2) size data structure, where k∈[1..n] is a parameter the user can choose, and ε>0 is an arbitrarily small constant, such that given any segment ab⎯⎯⎯⎯⎯⎯ and two points s,t∈P we can compute the Fréchet distance between ab⎯⎯⎯⎯⎯⎯ and the curve of P in between s and t in O((n/k)log2n+log4n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments. 
We also present two applications of our data structure: we show that we can compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n5/2+ε) time, and that we can efficiently find a translation of an arbitrary query segment ab⎯⎯⎯⎯⎯⎯ that minimizes the Fréchet distance with respect to a subcurve of P.
]]></description>
<dc:subject>computational-geometry computational-complexity algorithms distance preprocessing approximation rather-interesting to-simulate to-write-about consider:oracles</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:5471ba660ae6/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:distance"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:preprocessing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:oracles"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2106.04900">
    <title>[2106.04900] Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks</title>
    <dc:date>2022-07-09T12:40:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2106.04900</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Continuum mechanics simulators, numerically solving one or more partial differential equations, are essential tools in many areas of science and engineering, but their performance often limits application in practice. Recent modern machine learning approaches have demonstrated their ability to accelerate spatio-temporal predictions, although, with only moderate accuracy in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph neural network model for learning to infer unsteady continuum mechanics. MultiScaleGNN represents the physical domain as an unstructured set of nodes, and it constructs one or more graphs, each of them encoding different scales of spatial resolution. Successive learnt message passing between these graphs improves the ability of GNNs to capture and forecast the system state in problems encompassing a range of length scales. Using graph representations, MultiScaleGNN can impose periodic boundary conditions as an inductive bias on the edges in the graphs, and achieve independence to the nodes' positions. We demonstrate this method on advection problems and incompressible fluid dynamics. Our results show that the proposed model can generalise from uniform advection fields to high-gradient fields on complex domains at test time and infer long-term Navier-Stokes solutions within a range of Reynolds numbers. Simulations obtained with MultiScaleGNN are between two and four orders of magnitude faster than the ones on which it was trained.
]]></description>
<dc:subject>modeling approximation machine-learning neural-networks 20-years-ago-we-did-this-in-GP to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d2886cc4e68b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:modeling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:20-years-ago-we-did-this-in-GP"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1808.00572">
    <title>[1808.00572] Jumping champions and prime gaps using information-theoretic tools</title>
    <dc:date>2022-05-19T10:41:29+00:00</dc:date>
    <link>https://arxiv.org/abs/1808.00572</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study the spacing of the primes using methods from information theory. In information theory, the equivalence of continuous and discrete representations of information is established by Shannon sampling theory. Here, we use Shannon sampling methods to construct continuous functions whose varying bandwidth follows the distribution of the prime numbers. The Fourier transforms of these signals spike at frequently occurring spacings between the primes. We find prominent spikes, in particular, at the primorials. Previously, the primorials have been conjectured to be the most frequent gaps between subsequent primes, the so-called "jumping champions". Here, we find a foreshadowing of the primorial's role as jumping champions in the sense that Fourier spikes for the primorials arise much earlier on the number axis than where the primorials in question are expected to reign as jumping champions.
]]></description>
<dc:subject>number-theory primes approximation continuous-and-discrete-models rather-interesting signal-processing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0cadfb1287de/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:primes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continuous-and-discrete-models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:signal-processing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1801.07661">
    <title>[1801.07661] Approximability in the GPAC</title>
    <dc:date>2022-05-14T10:53:18+00:00</dc:date>
    <link>https://arxiv.org/abs/1801.07661</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Most of the physical processes arising in nature are modeled by either ordinary or partial differential equations. From the point of view of analog computability, the existence of an effective way to obtain solutions of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon as a model of the Differential Analyzer and improved by Pour-El, Lipshitz and Rubel, Costa and Graça and others. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, concerning the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. In particular, the Shannon GPAC cannot be used to generate non-differentially algebraic functions which can be approximately computed in other models of computation. We extend the class of data types using networks with channels which carry information on a general complete metric space X; for example X=C(R,R), the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and we add \emph{(continuous or discrete) limit} modules which have one input and one output. We then define an L-GPAC to be a network built with X-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. We study these analog systems and their associated operators, and show how some classically non-generable functions, such as the gamma function and the zeta function, can be captured with the L-GPAC.
]]></description>
<dc:subject>analog-computing representation approximation to-understand proof nonlinear-dynamics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8833ee7868c5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:analog-computing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2106.13914">
    <title>[2106.13914] Low-Precision Training in Logarithmic Number System using Multiplicative Weight Update</title>
    <dc:date>2022-04-30T21:50:09+00:00</dc:date>
    <link>https://arxiv.org/abs/2106.13914</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Representing deep neural networks (DNNs) in low-precision is a promising approach to enable efficient acceleration and memory reduction. Previous methods that train DNNs in low-precision typically keep a copy of weights in high-precision during the weight updates. Directly training with low-precision weights leads to accuracy degradation due to complex interactions between the low-precision number systems and the learning algorithms. To address this issue, we develop a co-designed low-precision training framework, termed LNS-Madam, in which we jointly design a logarithmic number system (LNS) and a multiplicative weight update algorithm (Madam). We prove that LNS-Madam results in low quantization error during weight updates, leading to a stable convergence even if the precision is limited. We further propose a hardware design of LNS-Madam that resolves practical challenges in implementing an efficient datapath for LNS computations. Our implementation effectively reduces energy overhead incurred by LNS-to-integer conversion and partial sum accumulation. Experimental results show that LNS-Madam achieves comparable accuracy to full-precision counterparts with only 8 bits on popular computer vision and natural language tasks. Compared to a full-precision floating-point implementation, LNS-Madam reduces the energy consumption by over 90.
]]></description>
<dc:subject>approximation neural-networks representation floating-point rather-interesting to-write-about to-simulate consider:lexicase consider:non-gradient-applications</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9c2dfdede684/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:floating-point"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:lexicase"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:non-gradient-applications"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2102.02365">
    <title>[2102.02365] Wind Field Reconstruction with Adaptive Random Fourier Features</title>
    <dc:date>2022-04-19T10:30:54+00:00</dc:date>
    <link>https://arxiv.org/abs/2102.02365</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging and Inverse distance weighting. Random Fourier features is a linear model β(xx)=∑Kk=1βkeiωkxx approximating the velocity field, with frequencies ωk randomly sampled and amplitudes βk trained to minimize a loss function. We include a physically motivated divergence penalty term |∇⋅β(xx)|2, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.
]]></description>
<dc:subject>approximation inverse-problems rather-interesting learning-from-data nonlinear-dynamics online-learning to-understand statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:3fa7289011de/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inverse-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:learning-from-data"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:online-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2105.07512">
    <title>[2105.07512] Substitutional Neural Image Compression</title>
    <dc:date>2022-03-17T10:52:01+00:00</dc:date>
    <link>https://arxiv.org/abs/2105.07512</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We describe Substitutional Neural Image Compression (SNIC), a general approach for enhancing any neural image compression model, that requires no data or additional tuning of the trained model. It boosts compression performance toward a flexible distortion metric and enables bit-rate control using a single model instance. The key idea is to replace the image to be compressed with a substitutional one that outperforms the original one in a desired way. Finding such a substitute is inherently difficult for conventional codecs, yet surprisingly favorable for neural compression models thanks to their fully differentiable structures. With gradients of a particular loss backpropogated to the input, a desired substitute can be efficiently crafted iteratively. We demonstrate the effectiveness of SNIC, when combined with various neural compression models and target metrics, in improving compression quality and performing bit-rate control measured by rate-distortion curves. Empirical results of control precision and generation speed are also discussed.
]]></description>
<dc:subject>image-processing neural-networks generative-models approximation compression to-write-about to-visualize consider:performance-measures machine-learning computational-complexity</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e52739abcbfa/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:image-processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:compression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1906.05746">
    <title>[1906.05746] Nonlinear System Identification via Tensor Completion</title>
    <dc:date>2022-03-04T11:21:16+00:00</dc:date>
    <link>https://arxiv.org/abs/1906.05746</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general nonlinear system, as they have proven to be very effective in approximating complex highly nonlinear functions. In this work, we show that identifying a general nonlinear function y=f(x1,…,xN) from input-output examples can be formulated as a tensor completion problem and under certain conditions provably correct nonlinear system identification is possible. Specifically, we model the interactions between the N input variables and the scalar output of a system by a single N-way tensor, and setup a weighted low-rank tensor completion problem with smoothness regularization which we tackle using a block coordinate descent algorithm. We extend our method to the multi-output setting and the case of partially observed data, which cannot be readily handled by neural networks. Finally, we demonstrate the effectiveness of the approach using several regression tasks including some standard benchmarks and a challenging student grade prediction task.
]]></description>
<dc:subject>approximation models-and-modes representation optimization system-identification tensors to-understand</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4cc8017689df/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:models-and-modes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:system-identification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tensors"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2110.14207">
    <title>[2110.14207] How Much Coffee Was Consumed During EMNLP 2019? Fermi Problems: A New Reasoning Challenge for AI</title>
    <dc:date>2022-02-21T09:23:15+00:00</dc:date>
    <link>https://arxiv.org/abs/2110.14207</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Many real-world problems require the combined application of multiple reasoning abilities employing suitable abstractions, commonsense knowledge, and creative synthesis of problem-solving strategies. To help advance AI systems towards such capabilities, we propose a new reasoning challenge, namely Fermi Problems (FPs), which are questions whose answers can only be approximately estimated because their precise computation is either impractical or impossible. For example, "How much would the sea level rise if all ice in the world melted?" FPs are commonly used in quizzes and interviews to bring out and evaluate the creative reasoning abilities of humans. To do the same for AI systems, we present two datasets: 1) A collection of 1k real-world FPs sourced from quizzes and olympiads; and 2) a bank of 10k synthetic FPs of intermediate complexity to serve as a sandbox for the harder real-world challenge. In addition to question answer pairs, the datasets contain detailed solutions in the form of an executable program and supporting facts, helping in supervision and evaluation of intermediate steps. We demonstrate that even extensively fine-tuned large scale language models perform poorly on these datasets, on average making estimates that are off by two orders of magnitude. Our contribution is thus the crystallization of several unsolved AI problems into a single, new challenge that we hope will spur further advances in building systems that can reason.
]]></description>
<dc:subject>natural-language-processing artificial-intelligence rather-interesting define-your-terms benchmarking approximation to-write-about consider:representation consider:metaheuristics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d5af6eaea474/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:natural-language-processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:artificial-intelligence"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:define-your-terms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:benchmarking"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:metaheuristics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1704.00362">
    <title>[1704.00362] Understanding Concept Drift</title>
    <dc:date>2022-02-13T12:37:09+00:00</dc:date>
    <link>https://arxiv.org/abs/1704.00362</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Concept drift is a major issue that greatly affects the accuracy and reliability of many real-world applications of machine learning. We argue that to tackle concept drift it is important to develop the capacity to describe and analyze it. We propose tools for this purpose, arguing for the importance of quantitative descriptions of drift in marginal distributions. We present quantitative drift analysis techniques along with methods for communicating their results. We demonstrate their effectiveness by application to three real-world learning tasks.
]]></description>
<dc:subject>machine-learning nonuniformity rather-interesting concept-drift approximation time-series to-write-about to-simulate consider:in-browser-data-sources</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e52f7be507d3/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonuniformity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:concept-drift"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:time-series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:in-browser-data-sources"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1803.00250">
    <title>[1803.00250] Distance Measure Machines</title>
    <dc:date>2022-02-12T13:26:09+00:00</dc:date>
    <link>https://arxiv.org/abs/1803.00250</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper presents a distance-based discriminative framework for learning with probability distributions. Instead of using kernel mean embeddings or generalized radial basis kernels, we introduce embeddings based on dissimilarity of distributions to some reference distributions denoted as templates. Our framework extends the theory of similarity of Balcan et al. (2008) to the population distribution case and we show that, for some learning problems, some dissimilarity on distribution achieves low-error linear decision functions with high probability. Our key result is to prove that the theory also holds for empirical distributions. Algorithmically, the proposed approach consists in computing a mapping based on pairwise dissimilarity where learning a linear decision function is amenable. Our experimental results show that the Wasserstein distance embedding performs better than kernel mean embeddings and computing Wasserstein distance is far more tractable than estimating pairwise Kullback-Leibler divergence of empirical distributions.
]]></description>
<dc:subject>machine-learning representation rather-interesting probability-theory to-understand approximation consider:EDAs consider:performance-measures</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:89b7794363b0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:EDAs"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2001.11692">
    <title>[2001.11692] Convolutional Embedding for Edit Distance</title>
    <dc:date>2022-02-08T12:29:44+00:00</dc:date>
    <link>https://arxiv.org/abs/2001.11692</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Edit-distance-based string similarity search has many applications such as spell correction, data de-duplication, and sequence alignment. However, computing edit distance is known to have high complexity, which makes string similarity search challenging for large datasets. In this paper, we propose a deep learning pipeline (called CNN-ED) that embeds edit distance into Euclidean distance for fast approximate similarity search. A convolutional neural network (CNN) is used to generate fixed-length vector embeddings for a dataset of strings and the loss function is a combination of the triplet loss and the approximation error. To justify our choice of using CNN instead of other structures (e.g., RNN) as the model, theoretical analysis is conducted to show that some basic operations in our CNN model preserve edit distance. Experimental results show that CNN-ED outperforms data-independent CGK embedding and RNN-based GRU embedding in terms of both accuracy and efficiency by a large margin. We also show that string similarity search can be significantly accelerated using CNN-based embeddings, sometimes by orders of magnitude.
]]></description>
<dc:subject>edit-distance neural-networks approximation heuristics rather-interesting to-understand nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:52092f6a70f1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:edit-distance"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:heuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2106.14274">
    <title>[2106.14274] Learning Mesh Representations via Binary Space Partitioning Tree Networks</title>
    <dc:date>2022-02-05T13:36:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2106.14274</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Polygonal meshes are ubiquitous, but have only played a relatively minor role in the deep learning revolution. State-of-the-art neural generative models for 3D shapes learn implicit functions and generate meshes via expensive iso-surfacing. We overcome these challenges by employing a classical spatial data structure from computer graphics, Binary Space Partitioning (BSP), to facilitate 3D learning. The core operation of BSP involves recursive subdivision of 3D space to obtain convex sets. By exploiting this property, we devise BSP-Net, a network that learns to represent a 3D shape via convex decomposition without supervision. The network is trained to reconstruct a shape using a set of convexes obtained from a BSP-tree built over a set of planes, where the planes and convexes are both defined by learned network weights. BSP-Net directly outputs polygonal meshes from the inferred convexes. The generated meshes are watertight, compact (i.e., low-poly), and well suited to represent sharp geometry. We show that the reconstruction quality by BSP-Net is competitive with those from state-of-the-art methods while using much fewer primitives. We also explore variations to BSP-Net including using a more generic decoder for reconstruction, more general primitives than planes, as well as training a generative model with variational auto-encoders. Code is available at this https URL.
]]></description>
<dc:subject>computational-geometry numerical-methods neural-networks machine-learning mesh-generation 3d optimization rather-interesting approximation to-write-about consider:performance-measures consider:edge-cases</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:7f9a89ab9655/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mesh-generation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:edge-cases"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2110.14151">
    <title>[2110.14151] Voxelization based packing analysis for discrete element simulations of non-spherical particles</title>
    <dc:date>2022-01-22T13:10:00+00:00</dc:date>
    <link>https://arxiv.org/abs/2110.14151</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A voxelization based post-processing algorithm is proposed to analyze the packing of non-spherical particle assemblies simulated using the Discrete Element Method. Voxelization of the particle data allows for isolating the geometric features of the granular assembly in various spatial sub-domains (2D surface or 3D region) and investigate the localized packing behaviour. Analyzing the local packing behaviour enables determining con-fined influences such as the wall-effect, stacking behaviour, local expansion/contraction and localized loading. The efficacy of the proposed technique to analyze practical granular assemblies is demonstrated through the packing analysis of different assemblies of superquadric cubes, superquadric ellipsoidals, and multi-spherical coffee beans.
]]></description>
<dc:subject>algorithms approximation numerical-methods 3d rather-interesting optimization to-write-about to-understand consider:visualization granular-materials</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:20ffc45d5548/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:granular-materials"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1910.11826">
    <title>[1910.11826] Weighted Quasi Interpolant Spline Approximations: Properties and Applications</title>
    <dc:date>2022-01-02T21:20:08+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.11826</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds we propose the Weighted Quasi Interpolant Spline Approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of nonparametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation and simulation of rainfall precipitations.
]]></description>
<dc:subject>numerical-methods optimization rather-interesting approximation algorithms to-understand</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9eb87d8d8377/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1511.02179">
    <title>[1511.02179] Irrational Base Counting</title>
    <dc:date>2022-01-01T13:29:33+00:00</dc:date>
    <link>https://arxiv.org/abs/1511.02179</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We will provide algorithmic implementation with proofs of existence and uniqueness for the Absolute and Alternating Ostrowski Numeration Systems.
]]></description>
<dc:subject>number-theory representation rather-interesting algorithms approximation to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:61f369c4e67d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1207.1067">
    <title>[1207.1067] Bounding differences in Jager Pairs</title>
    <dc:date>2022-01-01T13:22:30+00:00</dc:date>
    <link>https://arxiv.org/abs/1207.1067</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their difference. Results will also apply to the classical regular and backwards continued fractions expansions, which are realized as special cases.
]]></description>
<dc:subject>continued-fractions number-theory approximation rather-interesting dynamical-systems to-write-about to-simulate consider:stochastic-search</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a28f68ffe2fe/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:stochastic-search"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2106.10712">
    <title>[2106.10712] Solving for best linear approximates</title>
    <dc:date>2022-01-01T13:08:59+00:00</dc:date>
    <link>https://arxiv.org/abs/2106.10712</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Our goal is to finally settle a persistent problem in Diophantine Approximation, that of finding best inhomogeneous linear approximates. Classical results from the theory of continued fractions solve the special homogeneous case in the form of a complete sequence of normal approximates. Real expansions that allow the notion of normality to percolate into the inhomogeneous setting will provide us with the general solution.
]]></description>
<dc:subject>nonlinear-dynamics continued-fractions approximation rather-interesting number-theory to-write-about to-simulate consider:random-approximations optimization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f63fc99721fd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:random-approximations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2003.07798">
    <title>[2003.07798] Pressio: Enabling projection-based model reduction for large-scale nonlinear dynamical systems</title>
    <dc:date>2021-10-23T06:50:26+00:00</dc:date>
    <link>https://arxiv.org/abs/2003.07798</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This work introduces Pressio, an open-source project aimed at enabling leading-edge projection-based reduced order models (ROMs) for large-scale nonlinear dynamical systems in science and engineering. Pressio provides model-reduction methods that can reduce both the number of spatial and temporal degrees of freedom for any dynamical system expressible as a system of parameterized ordinary differential equations (ODEs). We leverage this simple, expressive mathematical framework as a pivotal design choice to enable a minimal application programming interface (API) that is natural to dynamical systems. The core component of Pressio is a C++11 header-only library that leverages generic programming to support applications with arbitrary data types and arbitrarily complex programming models. This is complemented with Python bindings to expose these C++ functionalities to Python users with negligible overhead and no user-required binding code. We discuss the distinguishing characteristics of Pressio relative to existing model-reduction libraries, outline its key design features, describe how the user interacts with it, and present two test cases -- including one with over 20 million degrees of freedom -- that highlight the performance results of Pressio and illustrate the breath of problems that can be addressed with it.
]]></description>
<dc:subject>nonlinear-dynamics approximation numerical-methods rather-interesting dynamical-systems to-understand system-identification</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:69ce0d534dcf/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:system-identification"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2003.02605">
    <title>[2003.02605] Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles</title>
    <dc:date>2021-10-23T06:47:31+00:00</dc:date>
    <link>https://arxiv.org/abs/2003.02605</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect. 
We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ϵ>0, assuming that the coordinates of all input objects are in [0,N]d and each of their edges has length at least 1. We obtain the following results: 
∙ For weighted intervals, we maintain a (1+ϵ)-approximate solution. 
∙ For d-dimensional hypercubes we maintain a (1+ϵ)2d-approximate solution in the unweighted case and a O(2d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1+ϵ)-approximate solution one needs polynomial update time for d≥2 if the ETH holds. 
∙ For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1+ϵ)logd−1N.
]]></description>
<dc:subject>computational-complexity computational-geometry multiobjective-optimization rather-interesting to-write-about to-simulate consider:genetic-programming approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4cb39f59b01a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:multiobjective-optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2109.11003">
    <title>[2109.11003] Rational approximations of irrational numbers</title>
    <dc:date>2021-10-06T11:00:16+00:00</dc:date>
    <link>https://arxiv.org/abs/2109.11003</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Given quantities Δ1,Δ2,⋯⩾0, a fundamental problem in Diophantine approximation is to understand which irrational numbers x have infinitely many reduced rational approximations a/q such that |x−a/q|<Δq. Depending on the choice of Δq and of x, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a "metric" point of view, the question is governed by a simple zero--one law: writing φ for Euler's totient function, we either have ∑∞q=1φ(q)Δq=∞ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or ∑∞q=1φ(q)Δq<∞ and almost no irrationals are approximable. We present the history of the Duffin--Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos--Maynard that settled it.
]]></description>
<dc:subject>number-theory approximation continued-fractions to-write-about to-simulate consider:constraint-satisfaction consider:locked-values</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:52d190d6ab3d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:locked-values"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1009.5202">
    <title>[1009.5202] Ramanujan-like series for $1/π^2$ and String Theory</title>
    <dc:date>2021-07-03T11:09:19+00:00</dc:date>
    <link>https://arxiv.org/abs/1009.5202</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for 1/π2 of hypergeometric and non-hypergeometric types.
]]></description>
<dc:subject>number-theory continued-fractions approximation rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:5e364145023f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.09715">
    <title>[2012.09715] Approximating inverse cumulative distribution functions to produce approximate random variables</title>
    <dc:date>2021-06-27T10:12:23+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.09715</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced by approximations to a distribution's inverse cumulative distribution function. These approximations are designed to be computationally inexpensive, and much cheaper than exact library functions, and thus highly suitable for use in Monte Carlo simulations. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals, and a piecewise linear using geometrically decaying intervals. The error of the approximations are bounded and the convergence demonstrated, and the computational savings measured for C and C++ implementations. Implementations tailored for Intel and Arm hardwares are inspected, alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speed ups without losing accuracy, offering speed ups by a factor of 5--7. These ideas are empirically extended to the Milstein scheme, and the Cox-Ingersoll-Ross process' non central chi-squared distribution, which offer speed ups by a factor of 250 or more.
]]></description>
<dc:subject>numerical-methods approximation rather-interesting simulation sampling performance-measure computational-complexity nudge-targets consider:representation consider:performance-measures consider:lexicase consider:extreme-values</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8ed61cde1044/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:performance-measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:lexicase"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:extreme-values"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.quantamagazine.org/how-rational-math-catches-slippery-irrational-numbers-20200310/">
    <title>How Rational Math Catches Slippery Irrational Numbers | Quanta Magazine</title>
    <dc:date>2021-06-18T13:49:19+00:00</dc:date>
    <link>https://www.quantamagazine.org/how-rational-math-catches-slippery-irrational-numbers-20200310/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Zooming in on irrational numbers in this way requires using bigger and bigger denominators: 1410, 141100, 1,4141,000, and so on. But a different approach, developed by the German mathematician Gustav Lejeune Dirichlet, generates good rational approximations using relatively small denominators. Dirichlet imagined the rational numbers spread out across the number line, and he figured out how to guarantee that every irrational number would be close to one. Let’s take a look at how he did it.]]></description>
<dc:subject>approximation number-theory algorithms nudge-targets consider:genetic-programming consider:worst-cases consider:representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:371c7e446424/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:worst-cases"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1802.03482">
    <title>[1802.03482] A Continuation Method for Discrete Optimization and its Application to Nearest Neighbor Classification</title>
    <dc:date>2021-06-07T10:52:32+00:00</dc:date>
    <link>https://arxiv.org/abs/1802.03482</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The continuation method is a popular approach in non-convex optimization and computer vision. The main idea is to start from a simple function that can be minimized efficiently, and gradually transform it to the more complicated original objective function. The solution of the simpler problem is used as the starting point to solve the original problem. We show a continuation method for discrete optimization problems. Ideally, we would like the evolved function to be hill-climbing friendly and to have the same global minima as the original function. We show that the proposed continuation method is the best affine approximation of a transformation that is guaranteed to transform the function to a hill-climbing friendly function and to have the same global minima. 
We show the effectiveness of the proposed technique in the problem of nearest neighbor classification. Although nearest neighbor methods are often competitive in terms of sample efficiency, the computational complexity in the test phase has been a major obstacle in their applicability in big data problems. Using the proposed continuation method, we show an improved graph-based nearest neighbor algorithm. The method is readily understood and easy to implement. We show how the computational complexity of the method in the test phase scales gracefully with the size of the training set, a property that is particularly important in big data applications.
]]></description>
<dc:subject>machine-learning heuristics approximation proxy-problems rather-interesting discrete-mathematics to-write-about consider:visualization consider:feature-discovery</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d91e1255bca1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:heuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proxy-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:discrete-mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.05503">
    <title>[2012.05503] Geometric algorithms for sampling the flux space of metabolic networks</title>
    <dc:date>2021-05-22T21:50:23+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.05503</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Systems Biology is a fundamental field and paradigm that introduces a new era in Biology. The crux of its functionality and usefulness relies on metabolic networks that model the reactions occurring inside an organism and provide the means to understand the underlying mechanisms that govern biological systems. Even more, metabolic networks have a broader impact that ranges from resolution of ecosystems to personalized medicine.The analysis of metabolic networks is a computational geometry oriented field as one of the main operations they depend on is sampling uniformly points from polytopes; the latter provides a representation of the steady states of the metabolic networks. However, the polytopes that result from biological data are of very high dimension (to the order of thousands) and in most, if not all, the cases are considerably skinny. Therefore, to perform uniform random sampling efficiently in this setting, we need a novel algorithmic and computational framework specially tailored for the properties of metabolic networks.We present a complete software framework to handle sampling in metabolic networks. Its backbone is a Multiphase Monte Carlo Sampling (MMCS) algorithm that unifies rounding and sampling in one pass, obtaining both upon termination. It exploits an improved variant of the Billiard Walk that enjoys faster arithmetic complexity per step. We demonstrate the efficiency of our approach by performing extensive experiments on various metabolic networks. Notably, sampling on the most complicated human metabolic network accessible today, Recon3D, corresponding to a polytope of dimension 5 335 took less than 30 hours. To our knowledge, that is out of reach for existing software.
]]></description>
<dc:subject>dynamical-systems reaction-networks simulation algorithms approximation sampling rather-interesting to-write-about consider:general-case consider:ReQ systems-biology</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:59ee2e9eec9c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:reaction-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:general-case"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ReQ"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:systems-biology"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2102.01161">
    <title>[2102.01161] Adjoint Rigid Transform Network: Task-conditioned Alignment of 3D Shapes</title>
    <dc:date>2021-05-19T11:29:08+00:00</dc:date>
    <link>https://arxiv.org/abs/2102.01161</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Most learning methods for 3D data (point clouds, meshes) suffer significant performance drops when the data is not carefully aligned to a canonical orientation. Aligning real world 3D data collected from different sources is non-trivial and requires manual intervention. In this paper, we propose the Adjoint Rigid Transform (ART) Network, a neural module which can be integrated with a variety of 3D networks to significantly boost their performance. ART learns to rotate input shapes to a learned canonical orientation, which is crucial for a lot of tasks such as shape reconstruction, interpolation, non-rigid registration, and latent disentanglement. ART achieves this with self-supervision and a rotation equivariance constraint on predicted rotations. The remarkable result is that with only self-supervision, ART facilitates learning a unique canonical orientation for both rigid and nonrigid shapes, which leads to a notable boost in performance of aforementioned tasks. We will release our code and pre-trained models for further research.
]]></description>
<dc:subject>computer-vision optimization approximation data-fusion rather-interesting to-write-about to-simulate consider:conflict-areas consider:anomaly-detection machine-learning neural-networks</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:08b400e783a4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computer-vision"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:data-fusion"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:conflict-areas"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:anomaly-detection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2103.01614">
    <title>[2103.01614] VEM and the Mesh</title>
    <dc:date>2021-03-28T12:20:56+00:00</dc:date>
    <link>https://arxiv.org/abs/2103.01614</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this work we report some results, obtained within the framework of the ERC Project CHANGE, on the impact on the performance of the virtual element method of the shape of the polygonal elements of the underlying mesh. More in detail, after reviewing the state of the art, we present a) an experimental analysis of the convergence of the VEM under condition violating the standard shape regularity assumptions, b) an analysis of the correlation between some mesh quality metrics and a set of different performance indexes, and c) a suitably designed mesh quality indicator, aimed at predicting the quality of the performance of the VEM on a given mesh.
]]></description>
<dc:subject>numerical-methods rather-interesting finite-elements-analysis algorithms computational-geometry approximation to-understand consider:performance-measures consider:visualization how-hard-to-show-in-browser?</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4e5f1293d09d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:finite-elements-analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:how-hard-to-show-in-browser?"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2009.00295">
    <title>[2009.00295] Predictive Accuracy of Wall-Modelled Large-Eddy Simulation on Unstructured Grids</title>
    <dc:date>2020-11-29T15:21:01+00:00</dc:date>
    <link>https://arxiv.org/abs/2009.00295</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The predictive accuracy of wall-modelled LES is influenced by a combination of the subgrid model, the wall model, the numerical dissipation induced primarily by the convective numerical scheme, and also by the density and topology of the computational grid. The latter factor is of particular importance for industrial flow problems, where unstructured grids are typically employed due to the necessity to handle complex geometries. Here, a systematic simulation-based study is presented, investigating the effect of grid-cell type on the predictive accuracy of wall-modelled LES in the framework of a general-purpose finite-volume solver. Following standard practice for meshing near-wall regions, it is proposed to use prismatic cells. Three candidate shapes for the base of the prisms are considered: a triangle, a quadrilateral, and an arbitrary polygon. The cell-centre distance is proposed as a metric to determine the spatial resolution of grids with different cell types. The simulation campaign covers two test cases with attached boundary layers: fully-developed turbulent channel flow, and a zero-pressure-gradient flat-plate turbulent boundary layer. A grid construction strategy is employed, which adapts the grid metric to the outer length scale of the boundary layer. The results are compared with DNS data concerning mean wall shear stress and profiles of flow statistics. The principle outcome is that unstructured simulations may provide the same accuracy as simulations on structured orthogonal hexahedral grids. The choice of base shape of the near-wall cells has a significant impact on the computational cost, but in terms of accuracy appears to be a factor of secondary importance.
]]></description>
<dc:subject>simulation fluid-dynamics representation looking-to-see rather-interesting modeling-is-not-mathematics to-write-about consider:simulation approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a4de464b0f00/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fluid-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:modeling-is-not-mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2010.11406">
    <title>[2010.11406] 1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two</title>
    <dc:date>2020-11-15T11:40:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2010.11406</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{generalized inverse} of a real matrix A is a matrix H that satisfies the Moore-Penrose (M-P) property AHA=A. If H also satisfies the M-P property HAH=H, then it is called \emph{reflexive}. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various \emph{sparse} reflexive generalized inverses of A. As is common, we use (vector) 1-norm minimization for both inducing sparsity and for keeping the magnitude of entries under control. 
When A is symmetric, a symmetric H is highly desirable, but generally such a restriction on H will not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. Letting the rank of A be r, we establish that the 1-norm minimizing generalized inverse of this type is a 1-norm minimizing symmetric generalized inverse when (i) r=1 and when (ii) r=2 and A is nonnegative. 
Another aspect of symmetry that we consider relates to another M-P property: H is \emph{ah-symmetric} if AH is symmetric. The ah-symmetry property is sufficient for a generalized inverse to be used to solve the least-squares problem min{‖Ax−b‖2: x∈ℝn} using H, via x:=Hb. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We establish that the 1-norm minimizing ah-symmetric generalized inverse of this type is a 1-norm minimizing ah-symmetric generalized inverse when (i) r=1 and when (ii) r=2 and A satisfies a technical condition.
]]></description>
<dc:subject>matrices inverse-problems rather-interesting approximation relaxations-of-rules to-write-about consider:looking-to-see consider:metaheuristics constraint-satisfaction multiobjective-optimization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ae9cc8c70f6c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:matrices"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inverse-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:relaxations-of-rules"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:metaheuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:multiobjective-optimization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1907.00205">
    <title>[1907.00205] The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants</title>
    <dc:date>2020-08-05T17:38:53+00:00</dc:date>
    <link>https://arxiv.org/abs/1907.00205</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Fundamental mathematical constants like e and π are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and usually discovered sporadically. Here we propose a novel and systematic approach that leverages algorithms for deriving mathematical formulas for fundamental constants and help reveal their underlying structure. Our algorithms find dozens of well-known as well as previously unknown continued fraction representations of π, e, Catalan's constant, and values of the Riemann zeta function. Two example conjectures found by our algorithm and so far unproven are:
24π2=2+7⋅0⋅1+8⋅142+7⋅1⋅2+8⋅242+7⋅2⋅3+8⋅342+7⋅3⋅4+8⋅44..,87ζ(3)=1⋅1−163⋅7−265⋅19−367⋅37−46..
We present two algorithms that proved useful in finding conjectures: a Meet-In-The-Middle (MITM) algorithm and a Gradient Descent (GD) tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values and thus they conjecture formulas without providing proofs and without requiring prior knowledge on any underlying mathematical structure. This approach is especially attractive for constants for which no mathematical structure is known, as it reverses the conventional approach of sequential logic in formal proofs. Instead, our work supports a different approach for research: algorithms utilizing numerical data to unveil mathematical structures, thus trying to play the role of intuition of great mathematicians of the past, providing leads to new mathematical research.
]]></description>
<dc:subject>continued-fractions metaheuristics approximation rather-interesting experimental-mathematics proof-systems to-write-about to-simulate consider:representation consider:performance-measures consider:genetic-programming</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1af9f125d57f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:metaheuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:experimental-mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1908.09791">
    <title>[1908.09791] Once-for-All: Train One Network and Specialize it for Efficient Deployment</title>
    <dc:date>2020-07-13T13:32:58+00:00</dc:date>
    <link>https://arxiv.org/abs/1908.09791</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We address the challenging problem of efficient inference across many devices and resource constraints, especially on edge devices. Conventional approaches either manually design or use neural architecture search (NAS) to find a specialized neural network and train it from scratch for each case, which is computationally prohibitive (causing CO2 emission as much as 5 cars' lifetime) thus unscalable. In this work, we propose to train a once-for-all (OFA) network that supports diverse architectural settings by decoupling training and search, to reduce the cost. We can quickly get a specialized sub-network by selecting from the OFA network without additional training. To efficiently train OFA networks, we also propose a novel progressive shrinking algorithm, a generalized pruning method that reduces the model size across many more dimensions than pruning (depth, width, kernel size, and resolution). It can obtain a surprisingly large number of sub-networks (>1019) that can fit different hardware platforms and latency constraints while maintaining the same level of accuracy as training independently. On diverse edge devices, OFA consistently outperforms state-of-the-art (SOTA) NAS methods (up to 4.0% ImageNet top1 accuracy improvement over MobileNetV3, or same accuracy but 1.5x faster than MobileNetV3, 2.6x faster than EfficientNet w.r.t measured latency) while reducing many orders of magnitude GPU hours and CO2 emission. In particular, OFA achieves a new SOTA 80.0% ImageNet top-1 accuracy under the mobile setting (<600M MACs). OFA is the winning solution for the 3rd Low Power Computer Vision Challenge (LPCVC), DSP classification track and the 4th LPCVC, both classification track and detection track. Code and 50 pre-trained models (for many devices & many latency constraints) are released at this https URL.
]]></description>
<dc:subject>machine-learning generalization sustainability rather-interesting to-write-about to-do consider:genetic-programming consider:once-and-for-all-GP transfer-learning approximation consider:performance-measures</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:fa59c5daa6a0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generalization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sustainability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-do"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:once-and-for-all-GP"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:transfer-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.03152">
    <title>[1909.03152] Graph Spanners: A Tutorial Review</title>
    <dc:date>2020-05-26T11:36:10+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.03152</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. It reviews the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.
]]></description>
<dc:subject>tutorial graph-theory approximation heuristics rather-interesting to-write-about to-simulate consider:random-guesses consider:relaxation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9e7af9693e7f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tutorial"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:heuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:random-guesses"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:relaxation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1803.04221">
    <title>[1803.04221] Extremal dependence of random scale constructions</title>
    <dc:date>2020-05-18T21:18:32+00:00</dc:date>
    <link>https://arxiv.org/abs/1803.04221</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2)=R(W1,W2), with non-degenerate R>0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.
]]></description>
<dc:subject>probability-theory representation bivariate-distributions correlation simulation approximation rather-interesting to-understand to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:98f5d851db63/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:bivariate-distributions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:correlation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
</rdf:RDF>