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    <description>recent bookmarks from cshalizi</description>
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    <title>now publishers - Universal Features for High-Dimensional Learning and Inference</title>
    <dc:date>2025-03-20T13:03:17+00:00</dc:date>
    <link>https://www.nowpublishers.com/article/Details/CIT-107</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This monograph develops unifying perspectives on the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, natural notions of universality are introduced, and a local equivalence among them is established. The analysis is naturally expressed via information geometry, which provides both conceptual and computational insights. The development reveals the complementary roles of the singular value decomposition, Hirschfeld-Gebelein-Rényi maximal correlation, the canonical correlation and principle component analyses of Hotelling and Pearson, Tishby’s information bottleneck, Wyner’s and Gács-Körner common information, Ky Fan k-norms, and Breiman and Friedman’s alternating conditional expectations algorithm. Among other uses, the framework facilitates understanding and optimizing aspects of learning systems, including multinomial logistic (softmax) regression and neural network architecture, matrix factorization methods for collaborative filtering and other applications, rank-constrained multivariate linear regression, and forms of semi-supervised learning."]]></description>
<dc:subject>to:NB information_theory pattern_discovery information_bottleneck principal_components low-dimensional_summaries information_geometry collaborative_filtering</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:62fffa70f391/</dc:identifier>
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<item rdf:about="https://www.countbayesie.com/blog/2023/4/21/linear-diffusion">
    <title>Linear Diffusion: Building a Diffusion Model from linear Components — Count Bayesie</title>
    <dc:date>2024-11-28T00:35:00+00:00</dc:date>
    <link>https://www.countbayesie.com/blog/2023/4/21/linear-diffusion</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[--- This is awesome.  It doesn't work that great, but it's awesome.  I'm tempted to have The Kids duplicate it with the eigendresses assignments in data mining, but unfortunately I don't think I saved the text when I scraped the images.]]></description>
<dc:subject>have_read data_mining principal_components generative_diffusion_models in_NB to_teach:statistics_and_generative_ai</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6ea3912b2b43/</dc:identifier>
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<item rdf:about="https://nowpublishers.com/article/Details/MAL-090">
    <title>now publishers - Minimum-Distortion Embedding</title>
    <dc:date>2021-10-18T13:50:48+00:00</dc:date>
    <link>https://nowpublishers.com/article/Details/MAL-090</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider the vector embedding problem. We are given a finite set of items, with the goal of assigning a representative vector to each one, possibly under some constraints (such as the collection of vectors being standardized, i.e., having zero mean and unit covariance). We are given data indicating that some pairs of items are similar, and optionally, some other pairs are dissimilar. For pairs of similar items, we want the corresponding vectors to be near each other, and for dissimilar pairs, we want the vectors to not be near each other, measured in Euclidean distance. We formalize this by introducing distortion functions, defined for some pairs of items. Our goal is to choose an embedding that minimizes the total distortion, subject to the constraints. We call this the minimum-distortion embedding (MDE) problem.
"The MDE framework is simple but general. It includes a wide variety of specific embedding methods, such as spectral embedding, principal component analysis, multidimensional scaling, Euclidean distance problems, dimensionality reduction methods (like Isomap and UMAP), semi-supervised learning, sphere packing, force-directed layout, and others. It also includes new embeddings, and provides principled ways of validating or sanity-checking historical and new embeddings alike.
"In a few special cases, MDE problems can be solved exactly. For others, we develop a projected quasi-Newton method that approximately minimizes the distortion and scales to very large data sets, while placing few assumptions on the distortion functions and constraints. This monograph is accompanied by an open-source Python package, PyMDE, for approximately solving MDE problems. Users can select from a library of distortion functions and constraints or specify custom ones, making it easy to rapidly experiment with new embeddings. Because our algorithm is scalable, and because PyMDE can exploit GPUs, our software scales to problems with millions of items and tens of millions of distortion functions. Additionally, PyMDE is competitive in runtime with specialized implementations of specific embedding methods. To demonstrate our method, we compute embeddings for several real-world data sets, including images, an academic co-author network, US county demographic data, and single-cell mRNA transcriptomes."
]]></description>
<dc:subject>to:NB to_read dimension_reduction principal_components graph_embedding re:hyperbolic_networks</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:828ed2b065ef/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/1809.08771">
    <title>[1809.08771] Modeling longitudinal data using matrix completion</title>
    <dc:date>2021-08-06T15:49:39+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.08771</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defective vision, or assessment of gait in patients with neurological disorders. Since the data collection is often costly and inconvenient, estimation of progression from sparse observations is of great interest for practitioners.
"From the statistical standpoint, such data is often analyzed in the context of a mixed-effect model where time is treated as both a fixed-effect (population progression curve) and a random-effect (individual variability). Alternatively, researchers analyze Gaussian processes or functional data where observations are assumed to be drawn from a certain distribution of processes. These models are flexible but rely on probabilistic assumptions, require very careful implementation, specific to the given problem, and tend to be slow in practice.
"In this study, we propose an alternative elementary framework for analyzing longitudinal data, relying on matrix completion. Our method yields estimates of progression curves by iterative application of the Singular Value Decomposition. Our framework covers multivariate longitudinal data, regression, and can be easily extended to other settings. As it relies on existing tools for matrix algebra it is efficient and easy to implement.
"We apply our methods to understand trends of progression of motor impairment in children with Cerebral Palsy. Our model approximates individual progression curves and explains 30% of the variability. Low-rank representation of progression trends enables identification of different progression trends in subtypes of Cerebral Palsy."]]></description>
<dc:subject>to:NB to_read time_series missing_data principal_components hastie.trevor to_teach:data_over_space_and_time statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:eac3788e7dd6/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2107.13894">
    <title>[2107.13894] Inference in heavy-tailed non-stationary multivariate time series</title>
    <dc:date>2021-07-30T02:55:07+00:00</dc:date>
    <link>https://arxiv.org/abs/2107.13894</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study inference on the common stochastic trends in a non-stationary, N-variate time series yt, in the possible presence of heavy tails. We propose a novel methodology which does not require any knowledge or estimation of the tail index, or even knowledge as to whether certain moments (such as the variance) exist or not, and develop an estimator of the number of stochastic trends m based on the eigenvalues of the sample second moment matrix of yt. We study the rates of such eigenvalues, showing that the first m ones diverge, as the sample size T passes to infinity, at a rate faster by O(T) than the remaining N−m ones, irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according to whether the relevant eigenvalue belongs to the set of the first m eigenvalues or not. We then construct a randomised statistic based on this, using it as part of a sequential testing procedure, ensuring consistency of the resulting estimator of m. We also discuss an estimator of the common trends based on principal components and show that, up to a an invertible linear transformation, such estimator is consistent in the sense that the estimation error is of smaller order than the trend itself. Finally, we also consider the case in which we relax the standard assumption of \textit{i.i.d.} innovations, by allowing for heterogeneity of a very general form in the scale of the innovations. A Monte Carlo study shows that the proposed estimator for m performs particularly well, even in samples of small size. We complete the paper by presenting four illustrative applications covering commodity prices, interest rates data, long run PPP and cryptocurrency markets."]]></description>
<dc:subject>to:NB time_series heavy_tails principal_components non-stationarity statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:3d5a2458f453/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2106.14238">
    <title>[2106.14238] Interpretable Network Representation Learning with Principal Component Analysis</title>
    <dc:date>2021-06-30T02:50:06+00:00</dc:date>
    <link>https://arxiv.org/abs/2106.14238</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider the problem of interpretable network representation learning for samples of network-valued data. We propose the Principal Component Analysis for Networks (PCAN) algorithm to identify statistically meaningful low-dimensional representations of a network sample via subgraph count statistics. The PCAN procedure provides an interpretable framework for which one can readily visualize, explore, and formulate predictive models for network samples. We furthermore introduce a fast sampling-based algorithm, sPCAN, which is significantly more computationally efficient than its counterpart, but still enjoys advantages of interpretability. We investigate the relationship between these two methods and analyze their large-sample properties under the common regime where the sample of networks is a collection of kernel-based random graphs. We show that under this regime, the embeddings of the sPCAN method enjoy a central limit theorem and moreover that the population level embeddings of PCAN and sPCAN are equivalent. We assess PCAN's ability to visualize, cluster, and classify observations in network samples arising in nature, including functional connectivity network samples and dynamic networks describing the political co-voting habits of the U.S. Senate. Our analyses reveal that our proposed algorithm provides informative and discriminatory features describing the networks in each sample. The PCAN and sPCAN methods build on the current literature of network representation learning and set the stage for a new line of research in interpretable learning on network-valued data. Publicly available software for the PCAN and sPCAN methods are available at this https URL."]]></description>
<dc:subject>to:NB network_data_analysis principal_components to_read statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4951d6ca8b4e/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2105.08875">
    <title>[2105.08875] Statistical Optimality and Computational Efficiency of Nyström Kernel PCA</title>
    <dc:date>2021-05-30T20:41:37+00:00</dc:date>
    <link>https://arxiv.org/abs/2105.08875</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Kernel methods provide an elegant framework for developing nonlinear learning algorithms from simple linear methods. Though these methods have superior empirical performance in several real data applications, their usefulness is inhibited by the significant computational burden incurred in large sample situations. Various approximation schemes have been proposed in the literature to alleviate these computational issues, and the approximate kernel machines are shown to retain the empirical performance. However, the theoretical properties of these approximate kernel machines are less well understood. In this work, we theoretically study the trade-off between computational complexity and statistical accuracy in Nyström approximate kernel principal component analysis (KPCA), wherein we show that the Nyström approximate KPCA matches the statistical performance of (non-approximate) KPCA while remaining computationally beneficial. Additionally, we show that Nyström approximate KPCA outperforms the statistical behavior of another popular approximation scheme, the random feature approximation, when applied to KPCA."]]></description>
<dc:subject>to:NB kernel_methods principal_components computational_statistics statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2c11c0e72152/</dc:identifier>
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<item rdf:about="https://elifesciences.org/articles/64948">
    <title>Polygenic Scores: How well can we separate genetics from the environment? | eLife</title>
    <dc:date>2021-01-14T18:47:24+00:00</dc:date>
    <link>https://elifesciences.org/articles/64948</link>
    <dc:creator>cshalizi</dc:creator><dc:subject>to:NB human_genetics principal_components</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:15e691be4d20/</dc:identifier>
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<item rdf:about="https://elifesciences.org/articles/61548">
    <title>Demographic history mediates the effect of stratification on polygenic scores | eLife</title>
    <dc:date>2021-01-14T18:46:27+00:00</dc:date>
    <link>https://elifesciences.org/articles/61548</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Population stratification continues to bias the results of genome-wide association studies (GWAS). When these results are used to construct polygenic scores, even subtle biases can cumulatively lead to large errors. To study the effect of residual stratification, we simulated GWAS under realistic models of demographic history. We show that when population structure is recent, it cannot be corrected using principal components of common variants because they are uninformative about recent history. Consequently, polygenic scores are biased in that they recapitulate environmental structure. Principal components calculated from rare variants or identity-by-descent segments can correct this stratification for some types of environmental effects. While family-based studies are immune to stratification, the hybrid approach of ascertaining variants in GWAS but reestimating effect sizes in siblings reduces but does not eliminate stratification. We show that the effect of population stratification depends not only on allele frequencies and environmental structure but also on demographic history."]]></description>
<dc:subject>to:NB human_genetics principal_components re:g_paper</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:e67b4648fc6b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:human_genetics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:g_paper"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2101.01839">
    <title>[2101.01839] Generalized Stochastic Processes as Linear Transformations of White Noise</title>
    <dc:date>2021-01-07T21:48:55+00:00</dc:date>
    <link>https://arxiv.org/abs/2101.01839</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We show that any (real) generalized stochastic process over ℝd can be expressed as a linear transformation of a White Noise process over ℝd. The procedure is done by using the regularity theorem for tempered distributions to obtain a mean-square continuous stochastic process which is then expressed in a Karhunen-Loève expansion with respect to a convenient Hilbert space. This result also allows to conclude that any generalized stochastic process can be expressed as a series expansion of deterministic tempered distributions weighted by uncorrelated random variables with square-summable variances. A result specifying when a generalized stochastic process can be linearly transformed into a White Noise is also presented."]]></description>
<dc:subject>to:NB stochastic_processes principal_components</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:3bf2fb8a28d4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.09141">
    <title>[2012.09141] Change Detection: A functional analysis perspective</title>
    <dc:date>2020-12-17T15:28:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.09141</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We develop a new approach for detecting changes in the behavior of stochastic processes and random fields based on tensor product representations such as the Karhunen-Loève expansion. From the associated eigenspaces of the covariance operator a series of nested function spaces are constructed, allowing detection of signals lying in orthogonal subspaces. In particular this can succeed even if the stochastic behavior of the signal changes either in a global or local sense. A mathematical approach is developed to locate and measure sizes of extraneous components based on construction of multilevel nested subspaces. We show examples in ℝ and on a spherical domain 𝕊2. However, the method is flexible, allowing the detection of orthogonal signals on general topologies, including spatio-temporal domains."]]></description>
<dc:subject>to:NB change-point_problem anomaly_detection principal_components stochastic_processes statistical_inference_for_stochastic_processes statistics spatio-temporal_statistics spatial_statistics time_series hilbert_space</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:20d5ce90dc91/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:change-point_problem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:anomaly_detection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_inference_for_stochastic_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:spatio-temporal_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:spatial_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hilbert_space"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2006.07691">
    <title>[2006.07691] Synthetic Interventions</title>
    <dc:date>2020-12-15T13:00:55+00:00</dc:date>
    <link>https://arxiv.org/abs/2006.07691</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Consider a panel data setting with observations of N units over T time steps. Each of the N units undergoes exactly one of D interventions at time step T0, with 1≤T0<T, prior to which all units experience no intervention, i.e., control. We present a causal framework, synthetic interventions (SI), to estimate the counterfactual outcome of each unit under each of the D interventions, averaged over the post-intervention time period. We prove identification of the causal parameter of interest under a latent factor model across time, units, and interventions. We furnish an algorithm to estimate the causal parameter, which utilizes principal component regression (PCR) as a key subroutine. We argue that PCR implicitly de-noises the observations, which are corrupted by idiosyncratic measurement error, and thus advocate for its usage in panel data settings. Formally, we establish consistency and asymptotic normality of the estimated causal parameter. We then compare our assumptions and results with those in the synthetic control (SC) literature. In doing so, we establish identification and inference results for SC as well. We further introduce a novel hypothesis test, with provable guarantees, to validate when to use SI (and thereby SC). Empirically, we showcase the efficacy of the SI framework on synthetic and real-world data. Finally, we discuss connections between the SI causal framework and tensor estimation."]]></description>
<dc:subject>to:NB time_series principal_components factor_analysis causal_inference statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:166f91ae66b1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:factor_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:causal_inference"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1706.06296">
    <title>[1706.06296] Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off</title>
    <dc:date>2020-12-04T21:43:57+00:00</dc:date>
    <link>https://arxiv.org/abs/1706.06296</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Kernel methods are powerful learning methodologies that provide a simple way to construct nonlinear algorithms from linear ones. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistical consistency of most of these approximate kernel methods is not well understood except for kernel ridge regression wherein it has been shown that the random feature approximation is not only computationally efficient but also statistically consistent with a minimax optimal rate of convergence. In this paper, we investigate the efficacy of random feature approximation in the context of kernel principal component analysis (KPCA) by studying the trade-off between computational and statistical behaviors of approximate KPCA. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces. The analysis hinges on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a self-adjoint Hilbert-Schmidt operator-valued U-statistics, which is of independent interest."]]></description>
<dc:subject>dimension_reduction kernel_methods random_features principal_components in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:36ef988d8d25/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kernel_methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_features"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.biorxiv.org/content/10.1101/2020.07.20.212530v1">
    <title>Demographic history impacts stratification in polygenic scores | bioRxiv</title>
    <dc:date>2020-11-19T20:15:25+00:00</dc:date>
    <link>https://www.biorxiv.org/content/10.1101/2020.07.20.212530v1</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Large genome-wide association studies (GWAS) have identified many loci exhibiting small but statistically significant associations with complex traits and disease risk. However, control of population stratification continues to be a limiting factor, particularly when calculating polygenic scores where subtle biases can cumulatively lead to large errors. We simulated GWAS under realistic models of demographic history to study the effect of residual stratification in large GWAS. We show that when population structure is recent, it cannot be fully corrected using principal components based on common variants—the standard approach—because common variants are uninformative about recent demographic history. Consequently, polygenic scores calculated from such GWAS results are biased in that they recapitulate non-genetic environmental structure. Principal components calculated from rare variants or identity-by-descent segments largely correct for this structure if environmental effects are smooth. However, even these corrections are not effective for local or batch effects. While sibling-based association tests are immune to stratification, the hybrid approach of ascertaining variants in a standard GWAS and then re-estimating effect sizes in siblings reduces but does not eliminate bias. Finally, we show that rare variant burden tests are relatively robust to stratification. Our results demonstrate that the effect of population stratification on GWAS and polygenic scores depends not only on the frequencies of tested variants and the distribution of environmental effects but also on the demographic history of the population."]]></description>
<dc:subject>to:NB genomics statistics prediction principal_components re:g_paper historical_genetics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:cb27d39ae079/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:genomics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:prediction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:g_paper"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:historical_genetics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.ejs/1597197614">
    <title>Barigozzi , Cho : Consistent estimation of high-dimensional factor models when the factor number is over-estimated</title>
    <dc:date>2020-11-16T16:23:26+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.ejs/1597197614</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["A high-dimensional rr-factor model for an nn-dimensional vector time series is characterised by the presence of a large eigengap (increasing with nn) between the rr-th and the (r+1)(r+1)-th largest eigenvalues of the covariance matrix. Consequently, Principal Component (PC) analysis is the most popular estimation method for factor models and its consistency, when rr is correctly estimated, is well-established in the literature. However, popular factor number estimators often suffer from the lack of an obvious eigengap in empirical eigenvalues and tend to over-estimate rr due, for example, to the existence of non-pervasive factors affecting only a subset of the series. We show that the errors in the PC estimators resulting from the over-estimation of rr are non-negligible, which in turn lead to the violation of the conditions required for factor-based large covariance estimation. To remedy this, we propose new estimators of the factor model based on scaling the entries of the sample eigenvectors. We show both theoretically and numerically that the proposed estimators successfully control for the over-estimation error, and investigate their performance when applied to risk minimisation of a portfolio of financial time series."]]></description>
<dc:subject>time_series factor_analysis high-dimensional_statistics inference_to_latent_objects statistics principal_components in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6a8174061a91/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:factor_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:inference_to_latent_objects"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://projecteuclid.org/euclid.ejs/1578020612">
    <title>Jones , Artemiou , Li : On the predictive potential of kernel principal components</title>
    <dc:date>2020-11-16T16:19:36+00:00</dc:date>
    <link>https://projecteuclid.org/euclid.ejs/1578020612</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We give a probabilistic analysis of a phenomenon in statistics which, until recently, has not received a convincing explanation. This phenomenon is that the leading principal components tend to possess more predictive power for a response variable than lower-ranking ones despite the procedure being unsupervised. Our result, in its most general form, shows that the phenomenon goes far beyond the context of linear regression and classical principal components — if an arbitrary distribution for the predictor XX and an arbitrary conditional distribution for Y|XY|X are chosen then any measureable function g(Y)g(Y), subject to a mild condition, tends to be more correlated with the higher-ranking kernel principal components than with the lower-ranking ones. The “arbitrariness” is formulated in terms of unitary invariance then the tendency is explicitly quantified by exploring how unitary invariance relates to the Cauchy distribution. The most general results, for technical reasons, are shown for the case where the kernel space is finite dimensional. The occurency of this tendency in real world databases is also investigated to show that our results are consistent with observation."]]></description>
<dc:subject>to:NB to_read principal_components statistics prediction</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2b4141a9695d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:prediction"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1910.06517">
    <title>[1910.06517] Principal Component Projection and Regression in Nearly Linear Time through Asymmetric SVRG</title>
    <dc:date>2019-10-16T15:20:27+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.06517</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Given a data matrix A∈ℝn×d, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of A, are fundamental problems in machine learning, optimization, and numerical analysis. In this paper we provide the first algorithms that solve these problems in nearly linear time for fixed eigenvalue distribution and large n. This improves upon previous methods which have superlinear running times when both the number of top eigenvalues and inverse gap between eigenspaces is large. We achieve our results by applying rational approximations to reduce PCP and PCR to solving asymmetric linear systems which we solve by a variant of SVRG. We corroborate these findings with preliminary empirical experiments."]]></description>
<dc:subject>to:NB principal_components linear_regression computational_statistics statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:b4041061a276/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:linear_regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:computational_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.wired.com/story/the-style-maven-astrophysicists-of-silicon-valley/">
    <title>The Style Maven Astrophysicists of Silicon Valley | WIRED</title>
    <dc:date>2019-10-10T20:39:10+00:00</dc:date>
    <link>https://www.wired.com/story/the-style-maven-astrophysicists-of-silicon-valley/</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Understanding latent style involves other physics principles too. Moody’s team uses something called eigenvector decomposition, a concept from quantum mechanics, to tease apart the overlapping “notes” in an individual’s style, sort of like “plucking a guitar string and listening for the multiple notes overlayed.” "

--- Oh for crying out loud.  I like to think that this is the journalist's cluelessness, rather than the ex-physicist's.]]></description>
<dc:subject>have_read data_mining physics principal_components utter_stupidity singular_value_decomposition_rules_everything_around_me to_teach:data-mining fashion</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:5a5beb3353d4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:physics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:utter_stupidity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:singular_value_decomposition_rules_everything_around_me"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:fashion"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1905.11926">
    <title>[1905.11926] Network Deconvolution</title>
    <dc:date>2019-10-01T16:39:15+00:00</dc:date>
    <link>https://arxiv.org/abs/1905.11926</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Convolution is a central operation in Convolutional Neural Networks (CNNs), which applies a kernel to overlapping regions shifted across the image. However, because of the immense amount of correlations in real-world image data, convolutional kernels are in effect re-learning redundant data. In this work, we show that this redundancy has made neural network training challenging, and propose network deconvolution, a procedure which optimally removes pixel-wise and channel-wise correlations before the data is fed into each layer. Network deconvolution can be efficiently calculated at a fraction of the computation cost of a convolution layer. We also show that the deconvolution filters in the first layer of the network resemble the center-surround structure found in biological neurons in the visual regions of the brain. Filtering with such kernels results in a sparse representation, a desired property that has been missing in the training of neural networks. Learning from the sparse representation promotes faster convergence and superior results without the use of batch normalization. We apply our network deconvolution operation to 10 modern neural network models by replacing batch normailization within each. Our extensive experiments show the network deconvolution operation is able to deliver performance improvement in all cases on CIFAR-10, CIFAR-100, MNIST, Fashion-MNIST and ImageNet datasets."

--- If they're just doing PCA I might cry.

--- ETA after skimming: they're doing PCA.]]></description>
<dc:subject>to:NB neural_networks statistics principal_components karl_pearson_died_for_your_sins your_favorite_deep_neural_network_sucks to_read</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6d139c0cc67c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:neural_networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:karl_pearson_died_for_your_sins"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:your_favorite_deep_neural_network_sucks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.10787">
    <title>[1909.10787] High-probability bounds for the reconstruction error of PCA</title>
    <dc:date>2019-09-26T18:01:25+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.10787</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We identify principal component analysis (PCA) as an empirical risk minimization problem and prove error bounds that hold with high probability. More precisely, we derive upper bounds for the reconstruction error of PCA that can be expressed relative to the minimal reconstruction error. The significance of these bounds is shown for the cases of functional and kernel PCA. In such scenarios, the eigenvalues of the covariance operator often decay at a polynomial or nearly exponential rate. Our results yield that the reconstruction error of PCA achieves the same rate as the minimal reconstruction error."]]></description>
<dc:subject>to:NB principal_components learning_theory to_read</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:e2344a55e38e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.03681">
    <title>[1909.03681] Outlier Detection in High Dimensional Data</title>
    <dc:date>2019-09-15T14:24:42+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.03681</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["High-dimensional data poses unique challenges in outlier detection process. Most of the existing algorithms fail to properly address the issues stemming from a large number of features. In particular, outlier detection algorithms perform poorly on data set of small size with a large number of features. In this paper, we propose a novel outlier detection algorithm based on principal component analysis and kernel density estimation. The proposed method is designed to address the challenges of dealing with high-dimensional data by projecting the original data onto a smaller space and using the innate structure of the data to calculate anomaly scores for each data point. Numerical experiments on synthetic and real-life data show that our method performs well on high-dimensional data. In particular, the proposed method outperforms the benchmark methods as measured by the F1-score. Our method also produces better-than-average execution times compared to the benchmark methods."

--- Seems OK but ad hoc.  Might make a decent extension to the eigendresses assignment for data mining.]]></description>
<dc:subject>to:NB anomaly_detection density_estimation principal_components high-dimensional_statistics statistics to_teach:data-mining</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c4ca0118845f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:anomaly_detection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:density_estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.03093">
    <title>[1909.03093] Solving Interpretable Kernel Dimension Reduction</title>
    <dc:date>2019-09-11T15:16:43+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.03093</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Kernel dimensionality reduction (KDR) algorithms find a low dimensional representation of the original data by optimizing kernel dependency measures that are capable of capturing nonlinear relationships. The standard strategy is to first map the data into a high dimensional feature space using kernels prior to a projection onto a low dimensional space. While KDR methods can be easily solved by keeping the most dominant eigenvectors of the kernel matrix, its features are no longer easy to interpret. Alternatively, Interpretable KDR (IKDR) is different in that it projects onto a subspace \textit{before} the kernel feature mapping, therefore, the projection matrix can indicate how the original features linearly combine to form the new features. Unfortunately, the IKDR objective requires a non-convex manifold optimization that is difficult to solve and can no longer be solved by eigendecomposition. Recently, an efficient iterative spectral (eigendecomposition) method (ISM) has been proposed for this objective in the context of alternative clustering. However, ISM only provides theoretical guarantees for the Gaussian kernel. This greatly constrains ISM's usage since any kernel method using ISM is now limited to a single kernel. This work extends the theoretical guarantees of ISM to an entire family of kernels, thereby empowering ISM to solve any kernel method of the same objective. In identifying this family, we prove that each kernel within the family has a surrogate Φ matrix and the optimal projection is formed by its most dominant eigenvectors. With this extension, we establish how a wide range of IKDR applications across different learning paradigms can be solved by ISM. To support reproducible results, the source code is made publicly available on \url{this https URL}."

--- Last tag is dreamily aspirational.]]></description>
<dc:subject>to:NB kernel_methods dimension_reduction principal_components to_teach:data-mining</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:3222a6b145d7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kernel_methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://psyarxiv.com/6m4ts/">
    <title>PsyArXiv Preprints | A “Need for Chaos” and the Sharing of Hostile Political Rumors in Advanced Democracies</title>
    <dc:date>2019-09-08T01:07:54+00:00</dc:date>
    <link>https://psyarxiv.com/6m4ts/</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The circulation of hostile political rumors (including but not limited to false news and conspiracy theories) has gained prominence in public debates across advanced democracies. Here, we provide the first comprehensive assessment of the psychological syndrome that elicits motivations to share hostile political rumors among citizens of democratic societies. Against the notion that sharing occurs to help one mainstream political actor in the increasingly polarized electoral competition against other mainstream actors, we demonstrate that sharing motivations are associated with ‘chaotic’ motivations to “burn down” the entire established democratic ‘cosmos’. We show that this extreme discontent is associated with motivations to share hostile political rumors, not because such rumors are viewed to be true but because they are believed to mobilize the audience against disliked elites. We introduce an individual difference measure, the “Need for Chaos”, to measure these motivations and illuminate their social causes, linked to frustrated status-seeking. Finally, we show that chaotic motivations are surprisingly widespread within advanced democracies, having some hold in up to 40 percent of the American national population."

]]></description>
<dc:subject>to:NB us_politics principal_components psychometrics #include:my_usual_skepticism_about_this_kind_of_psychometrics to_teach:data-mining epidemiology_of_representations social_media natural_history_of_truthiness re:actually-dr-internet-is-the-name-of-the-monsters-creator color_me_skeptical</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:0128253782dd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:us_politics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:psychometrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:#include:my_usual_skepticism_about_this_kind_of_psychometrics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:epidemiology_of_representations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:social_media"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:natural_history_of_truthiness"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:actually-dr-internet-is-the-name-of-the-monsters-creator"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:color_me_skeptical"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1908.02029">
    <title>[1908.02029] Online Detection of Sparse Changes in High-Dimensional Data Streams Using Tailored Projections</title>
    <dc:date>2019-08-07T12:32:40+00:00</dc:date>
    <link>https://arxiv.org/abs/1908.02029</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["When applying principal component analysis (PCA) for dimension reduction, the most varying projections are usually used in order to retain most of the information. For the purpose of anomaly and change detection, however, the least varying projections are often the most important ones. In this article, we present a novel method that automatically tailors the choice of projections to monitor for sparse changes in the mean and/or covariance matrix of high-dimensional data. A subset of the least varying projections is almost always selected based on a criteria of the projection's sensitivity to changes. 
"Our focus is on online/sequential change detection, where the aim is to detect changes as quickly as possible, while controlling false alarms at a specified level. A combination of tailored PCA and a generalized log-likelihood monitoring procedure displays high efficiency in detecting even very sparse changes in the mean, variance and correlation. We demonstrate on real data that tailored PCA monitoring is efficient for sparse change detection also when the data streams are highly auto-correlated and non-normal. Notably, error control is achieved without a large validation set, which is needed in most existing methods."]]></description>
<dc:subject>to:NB anomaly_detection change-point_problem principal_components statistics data_mining</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:df424a0256a1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:anomaly_detection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:change-point_problem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_mining"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.063304">
    <title>Phys. Rev. E 99, 063304 (2019) - Deriving the order parameters of a spin-glass model using principal component analysis</title>
    <dc:date>2019-06-14T12:38:10+00:00</dc:date>
    <link>https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.063304</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We investigate the relationship between the order parameters of spin models and principal component analysis (PCA). PCA is applied to the spin configurations generated from the Boltzmann distribution for the cases of uniformly interacting and randomly interacting spin models, and the datasets obtained for various specific temperatures are analyzed. In the case of the uniformly interacting spin model, the first principal component is found to be equivalent to the magnetization in the ordered phase, which is the order parameter. For the randomly interacting spin model, we apply the analysis to datasets generated by the Sherrington-Kirkpatrick model. When PCA is performed under the assumption that the Hadamard product of the spin configuration is taken as a new dataset, it is found that the first principal component coincides with the spin-glass order parameter. By analytically treating the limit in which the number of datasets is infinite, it is shown that the first principal component agrees with the order parameter."

--- If this works, it puts the "finding phase transitions with deep neural networks" papers in a very different light, I think...]]></description>
<dc:subject>to:NB principal_components statistical_mechanics phase_transitions</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:09d313c48f94/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistical_mechanics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:phase_transitions"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1712.05630">
    <title>[1712.05630] Sparse principal component analysis via random projections</title>
    <dc:date>2018-09-13T16:40:16+00:00</dc:date>
    <link>https://arxiv.org/abs/1712.05630</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. Our theory provides great detail on the statistical and computational trade-off in our procedure, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite-sample performance."]]></description>
<dc:subject>to:NB principal_components sparsity random_projections statistics samworth.richard_j.</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6c614249e1dd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_projections"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:samworth.richard_j."/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://dx.doi.org/10.1037/h0071325">
    <title>Analysis of a complex of statistical variables into principal components.</title>
    <dc:date>2018-09-08T18:52:35+00:00</dc:date>
    <link>http://dx.doi.org/10.1037/h0071325</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The problem is stated in detail, a method of analysis is derived and its geometrical meaning shown, methods of solution are illustrated and certain derivative problems are discussed. (To be concluded in October issue.) "

--- In which Harold Hotelling re-invents principal components analysis, 32 years after Karl Pearson.  (Part 2: http://dx.doi.org/10.1037/h0070888)]]></description>
<dc:subject>to:NB have_read principal_components data_analysis hotelling.harold re:ADAfaEPoV</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2e89a74cb6f8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hotelling.harold"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:ADAfaEPoV"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.tandfonline.com/doi/abs/10.1080/14786440109462720">
    <title>On lines and planes of closest fit to systems of points in space (K. Pearson, 1901)</title>
    <dc:date>2018-09-08T18:49:55+00:00</dc:date>
    <link>https://www.tandfonline.com/doi/abs/10.1080/14786440109462720</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[In which Karl Pearson invents principal components analysis, with the entirely sensible objective of finding low-dimensional approximations to high-dimensional data.  (i.e., basically the way I teach it!)]]></description>
<dc:subject>to:NB principal_components data_analysis pearson.karl re:ADAfaEPoV have_read</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:79b5a2fdcf75/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:pearson.karl"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:ADAfaEPoV"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://link.springer.com/article/10.1007/s11203-018-9172-1">
    <title>Optimal dimension reduction for high-dimensional and functional time series | SpringerLink</title>
    <dc:date>2018-07-13T02:43:55+00:00</dc:date>
    <link>https://link.springer.com/article/10.1007/s11203-018-9172-1</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Dimension reduction techniques are at the core of the statistical analysis of high-dimensional and functional observations. Whether the data are vector- or function-valued, principal component techniques, in this context, play a central role. The success of principal components in the dimension reduction problem is explained by the fact that, for any   K≤pK≤p , the K first coefficients in the expansion of a p-dimensional random vector   XX  in terms of its principal components is providing the best linear K-dimensional summary of   XX  in the mean square sense. The same property holds true for a random function and its functional principal component expansion. This optimality feature, however, no longer holds true in a time series context: principal components and functional principal components, when the observations are serially dependent, are losing their optimal dimension reduction property to the so-called dynamic principal components introduced by Brillinger in 1981 in the vector case and, in the functional case, their functional extension proposed by Hörmann, Kidziński and Hallin in 2015."]]></description>
<dc:subject>to:NB dimension_reduction time_series statistics principal_components</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4d73961cf5ab/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.pnas.org/content/115/2/E144.abstract.html">
    <title>Quantitative historical analysis uncovers a single dimension of complexity that structures global variation in human social organization</title>
    <dc:date>2018-01-09T20:46:25+00:00</dc:date>
    <link>http://www.pnas.org/content/115/2/E144.abstract.html</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Do human societies from around the world exhibit similarities in the way that they are structured, and show commonalities in the ways that they have evolved? These are long-standing questions that have proven difficult to answer. To test between competing hypotheses, we constructed a massive repository of historical and archaeological information known as “Seshat: Global History Databank.” We systematically coded data on 414 societies from 30 regions around the world spanning the last 10,000 years. We were able to capture information on 51 variables reflecting nine characteristics of human societies, such as social scale, economy, features of governance, and information systems. Our analyses revealed that these different characteristics show strong relationships with each other and that a single principal component captures around three-quarters of the observed variation. Furthermore, we found that different characteristics of social complexity are highly predictable across different world regions. These results suggest that key aspects of social organization are functionally related and do indeed coevolve in predictable ways. Our findings highlight the power of the sciences and humanities working together to rigorously test hypotheses about general rules that may have shaped human history."

--- Contributed, so the last tag applies very forcefully.]]></description>
<dc:subject>to:NB to_read comparative_history complexity_measures principal_components to_teach:undergrad-ADA color_me_skeptical</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:fc1b60107b6e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:comparative_history"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:complexity_measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:color_me_skeptical"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.calvinmurdock.com/aca/">
    <title>Additive Component Analysis – Calvin Murdock</title>
    <dc:date>2017-08-08T22:11:50+00:00</dc:date>
    <link>http://www.calvinmurdock.com/aca/</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal component analysis (PCA) is one of the most versatile tools for unsupervised learning with applications ranging from dimensionality reduction to exploratory data analysis and visualization. While much effort has been devoted to encouraging meaningful representations through regularization (e.g. non-negativity or sparsity), underlying linearity assumptions can limit their effectiveness. To address this issue, we propose Additive Component Analysis (ACA), a novel nonlinear extension of PCA. Inspired by multivariate nonparametric regression with additive models, ACA fits a smooth manifold to data by learning an explicit mapping from a low-dimensional latent space to the input space, which trivially enables applications like denoising. Furthermore, ACA can be used as a drop-in replacement in many algorithms that use linear component analysis methods as a subroutine via the local tangent space of the learned manifold. Unlike many other nonlinear dimensionality reduction techniques, ACA can be efficiently applied to large datasets since it does not require computing pairwise similarities or storing training data during testing. Multiple ACA layers can also be composed and learned jointly with essentially the same procedure for improved representational power, demonstrating the encouraging potential of nonparametric deep learning. We evaluate ACA on a variety of datasets, showing improved robustness, reconstruction performance, and interpretability."]]></description>
<dc:subject>to:NB dimension_reduction manifold_learning additive_models principal_components statistics to_read re:ADAfaEPoV</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:f0476aa38803/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:manifold_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:additive_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:ADAfaEPoV"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.oneweirdkerneltrick.com/catbasis.pdf">
    <title>Cat Basis Pursuit</title>
    <dc:date>2016-02-24T20:40:33+00:00</dc:date>
    <link>http://www.oneweirdkerneltrick.com/catbasis.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[Will the statistical machine learning reading group meet on 1 April?]]></description>
<dc:subject>machine_learning cats funny:geeky principal_components sparsity</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:031d454cc51b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:machine_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:cats"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:funny:geeky"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://graceavery.com/principalcomponentanalysisandfashion/">
    <title>Eigenstyle</title>
    <dc:date>2015-08-19T16:20:12+00:00</dc:date>
    <link>http://graceavery.com/principalcomponentanalysisandfashion/</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA[This is a perfectly nice example.  So how sexist am I that I am not going to swap out the cars-and-trucks one in my chapter on PCA for this?  (I guess I should at least mention it.)

--- Python code at https://github.com/graceavery/Eigenstyle but apparently not the original set of images]]></description>
<dc:subject>data_mining principal_components fashion have_read to:blog to_teach:data-mining re:ADAfaEPoV via:absfac</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4f7450d5ea6c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:fashion"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:blog"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:ADAfaEPoV"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:via:absfac"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1407.4578">
    <title>[1407.4578] Maximal Autocorrelation Functions in Functional Data Analysis</title>
    <dc:date>2015-01-23T13:27:57+00:00</dc:date>
    <link>http://arxiv.org/abs/1407.4578</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper proposes a new factor rotation for the context of functional principal components analysis. This rotation seeks to re-represent a functional subspace in terms of directions of decreasing smoothness as represented by a generalized smoothing metric. The rotation can be implemented simply and we show on two examples that this rotation can improve the interpretability of the leading components."]]></description>
<dc:subject>have_skimmed functional_data_analysis principal_components factor_analysis smoothing statistics hooker.giles to_teach:undergrad-ADA in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:8e83dc7a7faf/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_skimmed"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:functional_data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:factor_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:smoothing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hooker.giles"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1411.4681">
    <title>[1411.4681] Functional Principal Components Analysis of Spatially Correlated Data</title>
    <dc:date>2015-01-20T02:35:13+00:00</dc:date>
    <link>http://arxiv.org/abs/1411.4681</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["This paper focuses on the analysis of spatially correlated functional data. The between-curve correlation is modeled by correlating functional principal component scores of the functional data. We propose a Spatial Principal Analysis by Conditional Expectation framework to explicitly estimate spatial correlations and reconstruct individual curves. This approach works even when the observed data per curve are sparse. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov(Xi(s),Xi(t)) and cross-covariance surface Cov(Xi(s),Xj(t)) at locations indexed by i and j. Then a anisotropy Mat\'ern spatial correlation model is fit to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Simulation studies and applications of empirical data show improvements in the curve reconstruction using our framework over the method where curves are assumed to be independent. In addition, we show that the asymptotic properties of estimates in uncorrelated case still hold in our case if 'mild' spatial correlation is assumed."]]></description>
<dc:subject>to:NB functional_data_analysis spatial_statistics principal_components statistics hooker.giles</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4cd24bb63924/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:functional_data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:spatial_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hooker.giles"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1410.8260">
    <title>[1410.8260] Selecting the number of principal components: estimation of the true rank of a noisy matrix</title>
    <dc:date>2014-12-02T01:14:12+00:00</dc:date>
    <link>http://arxiv.org/abs/1410.8260</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal component analysis (PCA) is a well-known tool in multivariate statistics. One big challenge in using the method is the choice of the number of components. In this paper, we propose an exact distribution-based method for this purpose: our approach is related to the adaptive regression framework of Taylor et al. (2013). Assuming Gaussian noise, we use the conditional distribution of the eigenvalues of a Wishart matrix as our test statistic, and derive exact hypothesis tests and confidence intervals for the true singular values. In simulation studies we find that our proposed method compares well to the proposal of Kritchman & Nadler (2008), which uses the asymptotic distribution of singular values based on the Tracy-Widom laws."]]></description>
<dc:subject>to:NB principal_components model_selection statistics tibshirani.robert</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:fc0cabb54bff/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:model_selection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:tibshirani.robert"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://repository.upenn.edu/marketing_papers/13/">
    <title>&quot;Derivation of theory by means of factor analysis or Tom Swift and his electric factor analysis machine&quot;</title>
    <dc:date>2014-08-28T13:25:44+00:00</dc:date>
    <link>http://repository.upenn.edu/marketing_papers/13/</link>
    <dc:creator>cshalizi</dc:creator><dc:subject>have_read factor_analysis statistics model_discovery re:g_paper principal_components in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6ae62e192388/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:factor_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:model_discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:g_paper"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1406.0873">
    <title>[1406.0873] Unifying linear dimensionality reduction</title>
    <dc:date>2014-07-12T00:26:03+00:00</dc:date>
    <link>http://arxiv.org/abs/1406.0873</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of interest, such as covariance, dynamical structure, correlation between data sets, input-output relationships, and margin between data classes. Methods have been developed with a variety of names and motivations in many fields, and perhaps as a result the deeper connections between all these methods have not been understood. Here we unify methods from this disparate literature as optimization programs over matrix manifolds. We discuss principal component analysis, factor analysis, linear multidimensional scaling, Fisher's linear discriminant analysis, canonical correlations analysis, maximum autocorrelation factors, slow feature analysis, undercomplete independent component analysis, linear regression, and more. This optimization framework helps elucidate some rarely discussed shortcomings of well-known methods, such as the suboptimality of certain eigenvector solutions. Modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver, which accepts as input data and an objective to be optimized, and returns, as output, an optimal low-dimensional projection of the data. This optimization framework further allows rapid development of novel variants of classical methods, which we demonstrate here by creating an orthogonal-projection canonical correlations analysis. More broadly, we suggest that our generic linear dimensionality reduction solver can move linear dimensionality reduction toward becoming a blackbox, objective-agnostic numerical technology."]]></description>
<dc:subject>data_analysis principal_components factor_analysis optimization statistics dimension_reduction ghahramani.zoubin in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6bae21f8af20/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:factor_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:ghahramani.zoubin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1404.0788">
    <title>[1404.0788] On the principal components of sample covariance matrices</title>
    <dc:date>2014-04-20T18:32:08+00:00</dc:date>
    <link>http://arxiv.org/abs/1404.0788</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We introduce a class of M×M sample covariance matrices  which subsumes and generalizes several previous models. The associated population covariance matrix Σ=𝔼 is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IM may depend on M in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of . We derive precise large deviation estimates on the generalized components ⟨w,ξi⟩ of the outlier and non-outlier eigenvectors ξi. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \emph{subcritical} spikes of Σ. 
"The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix , satisfying 𝔼=IM, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution."]]></description>
<dc:subject>random_matrices principal_components in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bf34974fb041/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_matrices"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://jmlr.org/proceedings/papers/v33/kulesza14.html">
    <title>Low-Rank Spectral Learning | AISTATS 2014 | JMLR W&amp;CP</title>
    <dc:date>2014-04-20T17:47:44+00:00</dc:date>
    <link>http://jmlr.org/proceedings/papers/v33/kulesza14.html</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Spectral learning methods have recently been proposed as alternatives to slow, non-convex optimization algorithms like EM for a variety of probabilistic models in which hidden information must be inferred by the learner. These methods are typically controlled by a rank hyperparameter that sets the complexity of the model; when the model rank matches the true rank of the process generating the data, the resulting predictions are provably consistent and admit finite sample convergence bounds. However, in practice we usually do not know the true rank, and, in any event, from a computational and statistical standpoint it is likely to be prohibitively large. It is therefore of great practical interest to understand the behavior of low-rank spectral learning, where the model rank is less than the true rank. Counterintuitively, we show that even when the singular values omitted by lowering the rank are arbitrarily small, the resulting prediction errors can in fact be arbitrarily large. We identify two distinct possible causes for this bad behavior, and illustrate them with simple examples. We then show that these two causes are essentially complete: assuming that they do not occur, we can prove that the prediction error is bounded in terms of the magnitudes of the omitted singular values. We argue that the assumptions necessary for this result are relatively realistic, making low-rank spectral learning a viable option for many applications."]]></description>
<dc:subject>to:NB spectral_methods low-rank_approximation statistics computational_statistics principal_components re:AoS_project singh.satinder_baveja</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:273a202b361f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:spectral_methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:low-rank_approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:computational_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:AoS_project"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:singh.satinder_baveja"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.tandfonline.com/doi/abs/10.1080/01621459.2013.849199#.Uztl8tx_Tuc">
    <title>Taylor &amp; Francis Online :: Principal Flows - Journal of the American Statistical Association - Volume 109, Issue 505</title>
    <dc:date>2014-04-02T01:54:34+00:00</dc:date>
    <link>http://www.tandfonline.com/doi/abs/10.1080/01621459.2013.849199#.Uztl8tx_Tuc</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We revisit the problem of extending the notion of principal component analysis (PCA) to multivariate datasets that satisfy nonlinear constraints, therefore lying on Riemannian manifolds. Our aim is to determine curves on the manifold that retain their canonical interpretability as principal components, while at the same time being flexible enough to capture nongeodesic forms of variation. We introduce the concept of a principal flow, a curve on the manifold passing through the mean of the data, and with the property that, at any point of the curve, the tangent velocity vector attempts to fit the first eigenvector of a tangent space PCA locally at that same point, subject to a smoothness constraint. That is, a particle flowing along the principal flow attempts to move along a path of maximal variation of the data, up to smoothness constraints. The rigorous definition of a principal flow is given by means of a Lagrangian variational problem, and its solution is reduced to an ODE problem via the Euler–Lagrange method. Conditions for existence and uniqueness are provided, and an algorithm is outlined for the numerical solution of the problem. Higher order principal flows are also defined. It is shown that global principal flows yield the usual principal components on a Euclidean space. By means of examples, it is illustrated that the principal flow is able to capture patterns of variation that can escape other manifold PCA methods."]]></description>
<dc:subject>to:NB dimension_reduction principal_components manifold_learning geometry statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:ad0f6432b1db/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:manifold_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1402.0119">
    <title>[1402.0119] Randomized Nonlinear Component Analysis</title>
    <dc:date>2014-03-10T01:44:49+00:00</dc:date>
    <link>http://arxiv.org/abs/1402.0119</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Classical techniques such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques only reveal linear relationships in data. Although nonlinear variants of PCA and CCA have been proposed, they are computationally prohibitive in the large scale. 
"In a separate strand of recent research, randomized methods have been proposed to construct features that help reveal nonlinear patterns in data. For basic tasks such as regression or classification, random features exhibit little or no loss in performance, while achieving dramatic savings in computational requirements. 
"In this paper we leverage randomness to design scalable new variants of nonlinear PCA and CCA; our ideas also extend to key multivariate analysis tools such as spectral clustering or LDA. We demonstrate our algorithms through experiments on real-world data, on which we compare against the state-of-the-art. Code in R implementing our methods is provided in the Appendix."

--- This looks _awesome_.]]></description>
<dc:subject>to:NB to_read principal_components dimension_reduction statistics to_teach:undergrad-ADA random_features</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:79b245b38a8f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:random_features"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/euclid.aos/1388545673">
    <title>Vu , Lei : Minimax sparse principal subspace estimation in high dimensions</title>
    <dc:date>2014-02-20T22:20:20+00:00</dc:date>
    <link>http://projecteuclid.org/euclid.aos/1388545673</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study sparse principal components analysis in high dimensions, where p (the number of variables) can be much larger than n (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary notions of ℓq subspace sparsity: row sparsity and column sparsity. We prove nonasymptotic lower and upper bounds on the minimax subspace estimation error for 0≤q≤1. The bounds are optimal for row sparse subspaces and nearly optimal for column sparse subspaces, they apply to general classes of covariance matrices, and they show that ℓq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions. Interestingly, the form of the rates matches known results for sparse regression when the effective noise variance is defined appropriately. Our proof employs a novel variational sinΘ theorem that may be useful in other regularized spectral estimation problems."]]></description>
<dc:subject>to:NB principal_components sparsity dimension_reduction kith_and_kin vu.vincent statistics to_read minimax high-dimensional_statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:ed0ddaddd49e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vu.vincent"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:minimax"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.cs.huji.ac.il/~werman/Papers/cmds.pdf">
    <title>The World is not always Flat or Learning Curved Manifolds</title>
    <dc:date>2014-02-19T23:44:28+00:00</dc:date>
    <link>http://www.cs.huji.ac.il/~werman/Papers/cmds.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Manifold learning and finding low-dimensional structure in data is an important task. Many algorithms for this purpose embed data in Euclidean space, an approach which is destined to fail on non-flat data. This paper presents a non-iterative algebraic method for embedding the data into hyperbolic and spherical spaces. We argue that these spaces are often better than Euclidean space in capturing the geometry of the data. The approach can be used to extend algorithms such as ISOMAP and SDE to the curved case. We also demonstrate the utility of these embeddings by showing how some of the standard clustering algorithms translate to these curved manifolds."]]></description>
<dc:subject>to_read hyperbolic_geometry manifold_learning multidimensional_scaling re:smoothing_adjacency_matrices principal_components in_NB graph_embedding re:hyperbolic_networks</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:bf83be8b8be0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:hyperbolic_geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:manifold_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:multidimensional_scaling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:smoothing_adjacency_matrices"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:graph_embedding"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:re:hyperbolic_networks"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://link.springer.com/article/10.1007/s10618-013-0317-y">
    <title>Subspace clustering of high-dimensional data: a predictive approach - Springer</title>
    <dc:date>2014-02-19T02:46:07+00:00</dc:date>
    <link>http://link.springer.com/article/10.1007/s10618-013-0317-y</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In several application domains, high-dimensional observations are collected and then analysed in search for naturally occurring data clusters which might provide further insights about the nature of the problem. In this paper we describe a new approach for partitioning such high-dimensional data. Our assumption is that, within each cluster, the data can be approximated well by a linear subspace estimated by means of a principal component analysis (PCA). The proposed algorithm, Predictive Subspace Clustering (PSC) partitions the data into clusters while simultaneously estimating cluster-wise PCA parameters. The algorithm minimises an objective function that depends upon a new measure of influence for PCA models. A penalised version of the algorithm is also described for carrying our simultaneous subspace clustering and variable selection. The convergence of PSC is discussed in detail, and extensive simulation results and comparisons to competing methods are presented. The comparative performance of PSC has been assessed on six real gene expression data sets for which PSC often provides state-of-art results."]]></description>
<dc:subject>to:NB low-rank_approximation principal_components clustering data_mining statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:e61894d2a58c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:low-rank_approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:clustering"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1401.6978">
    <title>[1401.6978] Sparsistency and Agnostic Inference in Sparse PCA</title>
    <dc:date>2014-02-03T20:25:20+00:00</dc:date>
    <link>http://arxiv.org/abs/1401.6978</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["The presence of a sparse "truth" has been a constant assumption in the theoretical analysis of sparse PCA and is often implicit in its methodological development. This naturally raises questions about the properties of sparse PCA methods and how they depend on the assumption of sparsity. Under what conditions can the relevant variables be selected consistently if the truth is assumed to be sparse? If the truth is not sparse, let alone unique, what can be said about the results of sparse PCA? We answer these questions by investigating the properties of the recently proposed Fantope projection and selection (FPS) method in the high dimensional setting. Our results provide general sufficient conditions for sparsistency of the FPS estimator. These conditions are weak and can hold in situations where other estimators are known to fail. On the other hand, without assuming sparsity or identifiability, we show that FPS provides a sparse, linear dimension-reducing transformation that is close to the best possible in terms of maximizing the predictive covariance."]]></description>
<dc:subject>to:NB principal_components sparsity dimension_reduction kith_and_kin statistics vu.vincent lei.jing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2a94c1eec2f6/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:vu.vincent"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:lei.jing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1312.1801">
    <title>[1312.1801] Visualizing genetic constraints</title>
    <dc:date>2014-01-02T18:35:18+00:00</dc:date>
    <link>http://arxiv.org/abs/1312.1801</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal Components Analysis (PCA) is a common way to study the sources of variation in a high-dimensional data set. Typically, the leading principal components are used to understand the variation in the data or to reduce the dimension of the data for subsequent analysis. The remaining principal components are ignored since they explain little of the variation in the data. However, evolutionary biologists gain important insights from these low variation directions. Specifically, they are interested in directions of low genetic variability that are biologically interpretable. These directions are called genetic constraints and indicate directions in which a trait cannot evolve through selection. Here, we propose studying the subspace spanned by low variance principal components by determining vectors in this subspace that are simplest. Our method and accompanying graphical displays enhance the biologist's ability to visualize the subspace and identify interpretable directions of low genetic variability that align with simple directions."

- How are directions of selective constraint separated from directions where relevant mutants just haven't arisen? For that matter, how are they separated from directions which are accidentally missing because n is vastly less than p?  (Say 500k SNPs vs. 500 subjects if very lucky.)  Unusually low variance for the selective directions?]]></description>
<dc:subject>to:NB genetics principal_components visual_display_of_quantitative_information</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:164317fa5ea2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:genetics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:visual_display_of_quantitative_information"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1211.2671">
    <title>[1211.2671] A General Framework For Consistency of Principal Component Analysis</title>
    <dc:date>2013-11-14T04:51:42+00:00</dc:date>
    <link>http://arxiv.org/abs/1211.2671</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["A general asymptotic framework is developed for studying consis- tency properties of principal component analysis (PCA). Our frame- work includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. We also establish the corresponding convergence rate within each region. Under general spike covariance models, the dimension (or the number of variables) discourages the consistency of PCA, while the sample size and spike information (the relative size of the population eigenvalues) encourages PCA consistency. Our framework nicely illustrates the relationship among these three types of information in terms of dimension, sample size and spike size, and rigorously characterizes how their relationships a?ffect PCA consistency."]]></description>
<dc:subject>to:NB principal_components statistics estimation high-dimensional_statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:c83fea219731/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1310.5089">
    <title>[1310.5089] Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods</title>
    <dc:date>2013-10-23T14:25:13+00:00</dc:date>
    <link>http://arxiv.org/abs/1310.5089</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Feature extraction and dimensionality reduction are important tasks in many fields of science dealing with signal processing and analysis. The relevance of these techniques is increasing as current sensory devices are developed with ever higher resolution, and problems involving multimodal data sources become more common. A plethora of feature extraction methods are available in the literature collectively grouped under the field of Multivariate Analysis (MVA). This paper provides a uniform treatment of several methods: Principal Component Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis (CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions derived by means of the theory of reproducing kernel Hilbert spaces. We also review their connections to other methods for classification and statistical dependence estimation, and introduce some recent developments to deal with the extreme cases of large-scale and low-sized problems. To illustrate the wide applicability of these methods in both classification and regression problems, we analyze their performance in a benchmark of publicly available data sets, and pay special attention to specific real applications involving audio processing for music genre prediction and hyperspectral satellite images for Earth and climate monitoring."]]></description>
<dc:subject>to:NB signal_processing dimension_reduction regression principal_components kernel_methods statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:a643506f78fb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:signal_processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kernel_methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1309.2895">
    <title>[1309.2895] Sparse and Functional Principal Components Analysis</title>
    <dc:date>2013-09-12T20:05:36+00:00</dc:date>
    <link>http://arxiv.org/abs/1309.2895</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Regularized principal components analysis, especially Sparse PCA and Functional PCA, has become widely used for dimension reduction in high-dimensional settings. Many examples of massive data, however, may benefit from estimating both sparse AND functional factors. These include neuroimaging data where there are discrete brain regions of activation (sparsity) but these regions tend to be smooth spatially (functional). Here, we introduce an optimization framework that can encourage both sparsity and smoothness of the row and/or column PCA factors. This framework generalizes many of the existing approaches to Sparse PCA, Functional PCA and two-way Sparse PCA and Functional PCA, as these are all special cases of our method. In particular, our method permits flexible combinations of sparsity and smoothness that lead to improvements in feature selection and signal recovery as well as more interpretable PCA factors. We demonstrate our method on simulated data and a neuroimaging example on EEG data. This work provides a unified framework for regularized PCA that can form the foundation for a cohesive approach to regularization in high-dimensional multivariate analysis."]]></description>
<dc:subject>to:NB sparsity functional_data_analysis principal_components statistics dimension_reduction allen.genevera_i.</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:def8a3768d6a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:functional_data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:allen.genevera_i."/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://jmlr.org/papers/v14/dhillon13a.html">
    <title>A Risk Comparison of Ordinary Least Squares vs Ridge Regression</title>
    <dc:date>2013-07-20T13:11:25+00:00</dc:date>
    <link>http://jmlr.org/papers/v14/dhillon13a.html</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a principal component analysis) and then performs an ordinary (un- regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method (PCA-OLS) is within a constant factor (namely 4) of the risk of ridge regression (RR)."

!!!]]></description>
<dc:subject>regression statistics learning_theory principal_components to_teach:undergrad-ADA kakade.sham have_read high-dimensional_statistics foster.dean_p. to_teach:data-mining in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:58ee6967d573/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:regression"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:learning_theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kakade.sham"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:foster.dean_p."/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1307.0164">
    <title>[1307.0164] Sparse Principal Component Analysis for High Dimensional Vector Autoregressive Models</title>
    <dc:date>2013-07-02T03:15:45+00:00</dc:date>
    <link>http://arxiv.org/abs/1307.0164</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transition matrix plays an essential role in determining the final rates. We also characterize sufficient conditions under which sparse principal component analysis attains the optimal parametric rate. Our theoretical results are backed up by thorough numerical studies."]]></description>
<dc:subject>to:NB time_series principal_components high-dimensional_statistics liu.han statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:237266a4fefe/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:liu.han"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aoas/1372338471">
    <title>Gaydos , Heckman , Kirkpatrick , Stinchcombe , Schmitt , Kingsolver , Marron : Visualizing genetic constraints</title>
    <dc:date>2013-06-27T15:13:12+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aoas/1372338471</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal Components Analysis (PCA) is a common way to study the sources of variation in a high-dimensional data set. Typically, the leading principal components are used to understand the variation in the data or to reduce the dimension of the data for subsequent analysis. The remaining principal components are ignored since they explain little of the variation in the data. However, evolutionary biologists gain important insights from these low variation directions. Specifically, they are interested in directions of low genetic variability that are biologically interpretable. These directions are called genetic constraints and indicate directions in which a trait cannot evolve through selection. Here, we propose studying the subspace spanned by low variance principal components by determining vectors in this subspace that are simplest. Our method and accompanying graphical displays enhance the biologist’s ability to visualize the subspace and identify interpretable directions of low genetic variability that align with simple directions."

--- Worth mentioning in the PCA chapter of ADAfaEPoV?]]></description>
<dc:subject>to:NB principal_components genetics visual_display_of_quantitative_information data_analysis to_teach:undergrad-ADA</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:76215827f891/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:genetics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:visual_display_of_quantitative_information"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1305.5646">
    <title>[1305.5646] A simple proof for the multivariate Chebyshev inequality</title>
    <dc:date>2013-05-27T12:07:39+00:00</dc:date>
    <link>http://arxiv.org/abs/1305.5646</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this paper a simple proof of the Chebyshev's inequality for random vectors obtained by Chen (arXiv:0707.0805v2, 2011) is obtained. This inequality gives a lower bound for the percentage of the population of an arbitrary random vector X with finite mean E(X) and a positive definite covariance matrix V=Cov(X) whose Mahalanobis distance with respect to V to the mean E(X) is less than a fixed value. The proof is based on the calculation of the principal components."

--- As he says, this proof is so elementary it could go in any textbook.  Neat.]]></description>
<dc:subject>probability principal_components have_read to_teach:undergrad-ADA in_NB deviation_inequalities</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:1dddb31a0283/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:probability"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:deviation_inequalities"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aos/1368018173">
    <title>Ma : Sparse principal component analysis and iterative thresholding</title>
    <dc:date>2013-05-08T14:10:30+00:00</dc:date>
    <link>http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.aos/1368018173</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of features p is comparable to, or even much larger than, the sample size n. In this paper, we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse. Under a spiked covariance model, we find that the new approach recovers the principal subspace and leading eigenvectors consistently, and even optimally, in a range of high-dimensional sparse settings. Simulated examples also demonstrate its competitive performance."]]></description>
<dc:subject>to:NB sparsity principal_components statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:88b46fa0663f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.jstatsoft.org/v53/i03">
    <title>Fast and Robust Bootstrap for Multivariate Inference: The R Package FRB</title>
    <dc:date>2013-04-22T17:18:58+00:00</dc:date>
    <link>http://www.jstatsoft.org/v53/i03</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We present the FRB package for R, which implements the fast and robust bootstrap. This method constitutes an alternative to ordinary bootstrap or asymptotic inference procedures when using robust estimators such as S-, MM- or GS-estimators. The package considers three multivariate settings: principal components analysis, Hotelling tests and multivariate regression. It provides both the robust point estimates and uncertainty measures based on the fast and robust bootstrap. In this paper we give some background on the method, discuss the implementation and provide various examples."]]></description>
<dc:subject>principal_components bootstrap statistics robust_statistics in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:4d0eb3e5de5c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bootstrap"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:robust_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1302.6102">
    <title>[1302.6102] Functional Data Analysis with Increasing Number of Projections</title>
    <dc:date>2013-03-06T15:38:08+00:00</dc:date>
    <link>http://arxiv.org/abs/1302.6102</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. 
"We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example."]]></description>
<dc:subject>to:NB principal_components functional_data_analysis statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:8c3b7f4dc4d9/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:functional_data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1302.2752">
    <title>[1302.2752] Adaptive Metric Dimensionality Reduction</title>
    <dc:date>2013-03-06T15:29:12+00:00</dc:date>
    <link>http://arxiv.org/abs/1302.2752</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We initiate the study of dimensionality reduction in general metric spaces in the context of supervised learning. Our statistical contribution consists of tight Rademacher bounds for Lipschitz functions in metric spaces that are doubling, or nearly doubling. As a by-product, we obtain a new theoretical explanation for the empirically reported improvements gained by pre-processing Euclidean data by PCA (Principal Components Analysis) prior to constructing a linear classifier. 
"On the algorithmic front, we describe an analogue of PCA for metric spaces, namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Thus, our approach can exploit the dual benefits of low dimensionality: (1) more efficient proximity search algorithms, and (2) more optimistic generalization bounds."]]></description>
<dc:subject>to:NB dimension_reduction principal_components machine_learning kith_and_kin kontorovich.aryeh</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:5968ffe22d74/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:machine_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kontorovich.aryeh"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1302.1232">
    <title>[1302.1232] When are the most informative components for inference also the principal components?</title>
    <dc:date>2013-03-06T14:59:24+00:00</dc:date>
    <link>http://arxiv.org/abs/1302.1232</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Which components of the singular value decomposition of a signal-plus-noise data matrix are most informative for the inferential task of detecting or estimating an embedded low-rank signal matrix? Principal component analysis ascribes greater importance to the components that capture the greatest variation, i.e., the singular vectors associated with the largest singular values. This choice is often justified by invoking the Eckart-Young theorem even though that work addresses the problem of how to best represent a signal-plus-noise matrix using a low-rank approximation and not how to best_infer_ the underlying low-rank signal component. 
"Here we take a first-principles approach in which we start with a signal-plus-noise data matrix and show how the spectrum of the noise-only component governs whether the principal or the middle components of the singular value decomposition of the data matrix will be the informative components for inference. Simply put, if the noise spectrum is supported on a connected interval, in a sense we make precise, then the use of the principal components is justified. When the noise spectrum is supported on multiple intervals, then the middle components might be more informative than the principal components. 
"The end result is a proper justification of the use of principal components in the setting where the noise matrix is i.i.d. Gaussian and the identification of scenarios, generically involving heterogeneous noise models such as mixtures of Gaussians, where the middle components might be more informative than the principal components so that they may be exploited to extract additional processing gain. Our results show how the blind use of principal components can lead to suboptimal or even faulty inference because of phase transitions that separate a regime where the principal components are informative from a regime where they are uninformative."]]></description>
<dc:subject>to:NB principal_components statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2848db85aac3/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1104.3472">
    <title>[1104.3472] Principal arc analysis on direct product manifolds</title>
    <dc:date>2013-02-21T23:22:14+00:00</dc:date>
    <link>http://arxiv.org/abs/1104.3472</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis."]]></description>
<dc:subject>to:NB statistics dimension_reduction principal_components statistics_on_manifolds</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2bf4e5566e74/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics_on_manifolds"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1111.4601">
    <title>[1111.4601] Non-Asymptotic Analysis of Tangent Space Perturbation</title>
    <dc:date>2013-02-21T23:19:29+00:00</dc:date>
    <link>http://arxiv.org/abs/1111.4601</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data samples from a nonlinear manifold, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. Using eigenspace perturbation theory and non-asymptotic random matrix theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals an appropriate scale for local tangent plane recovery. We also introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for stable recovery."]]></description>
<dc:subject>statistics manifold_learning principal_components in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2411b784b629/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:manifold_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1301.0554">
    <title>[1301.0554] Tree-dependent Component Analysis</title>
    <dc:date>2013-01-07T22:48:13+00:00</dc:date>
    <link>http://arxiv.org/abs/1301.0554</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. Treating the problem as a semiparametric statistical problem, we show that the optimal transform is found by minimizing a contrast function based on mutual information, a function that directly extends the contrast function used for classical ICA. We provide two approximations of this contrast function, one using kernel density estimation, and another using kernel generalized variance. This tree-dependent component analysis framework leads naturally to an efficient general multivariate density estimation technique where only bivariate density estimation needs to be performed."

--- How did I miss this?!?]]></description>
<dc:subject>to:NB dimension_reduction principal_components graphical_models jordan.michael_i. bach.francis_r. density_estimation chow-liu_trees machine_learning statistics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:467d041d3e7c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:graphical_models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:jordan.michael_i."/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:bach.francis_r."/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:density_estimation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:chow-liu_trees"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:machine_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.math.psu.edu/morton/publications/pcca.pdf">
    <title>Principal Cumulant Component Analysis (Jason Morton, Lek-Heng Lim) [pdf]</title>
    <dc:date>2013-01-03T01:45:05+00:00</dc:date>
    <link>http://www.math.psu.edu/morton/publications/pcca.pdf</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Multivariate Gaussian data is completely characterized by its mean and covariance, yet modern non-Gaussian data makes higher-order statistics such as cumulants inevitable. For univariate data, the third and fourth scalar-valued cumulants are relatively well-studied as skewness and kurtosis. For multivariate data, these cumulants are tensor-valued, higher-order analogs of the covariance matrix capturing higher-order dependence in the data. In addition to their relative obscurity, there are few effective methods for analyzing these cumulant tensors. We propose a technique along the lines of Principal Component Analysis and Independent Component Analysis to analyze multivariate, non-Gaussian data motivated by the multilinear algebraic properties of cumulants. Our method relies on finding principal cumulant components that account for most of the variation in all higher-order cumulants, just as PCA obtains varimax components. An efficient algorithm based on limited-memory quasi-Newton maximization over a Grassmannian, using only standard matrix operations, may be used to find the principal cumulant components. Numerical experiments include forecasting higher portfolio moments and image dimension reduction."

- The to_teach tags here mean "to mention as further reading".]]></description>
<dc:subject>to:NB have_read statistics data_analysis cumulants via:arsyed principal_components to_teach:data-mining to_teach:undergrad-ADA dimension_reduction</dc:subject>
<dc:identifier>https://pinboard.in/u:cshalizi/b:a056b61da10e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:have_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:cumulants"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:via:arsyed"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:data-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_teach:undergrad-ADA"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.mitpressjournals.org/doi/abs/10.1162/NECO_a_00382">
    <title>Point-Process Principal Components Analysis via Geometric Optimization</title>
    <dc:date>2012-12-08T14:16:08+00:00</dc:date>
    <link>http://www.mitpressjournals.org/doi/abs/10.1162/NECO_a_00382</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["There has been a fast-growing demand for analysis tools for multivariate point-process data driven by work in neural coding and, more recently, high-frequency finance. Here we develop a true or exact (as opposed to one based on time binning) principal components analysis for preliminary processing of multivariate point processes. We provide a maximum likelihood estimator, an algorithm for maximization involving steepest ascent on two Stiefel manifolds, and novel constrained asymptotic analysis. The method is illustrated with a simulation and compared with a binning approach."]]></description>
<dc:subject>point_processes time_series data_analysis principal_components solo.victor statistics in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:56cdd211d1e8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:point_processes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:time_series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:solo.victor"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1211.5803">
    <title>[1211.5803] Fast network community detection by SCORE</title>
    <dc:date>2012-11-28T14:24:50+00:00</dc:date>
    <link>http://arxiv.org/abs/1211.5803</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Consider a network where the nodes split into K different communities. The community labels for the nodes are unknown and it is of major interest to estimate them (i.e., community detection). Degree Corrected Block Model (DCBM) is a popular network model. How to detect communities with the DCBM is an interesting problem, where the main challenge lies in the degree heterogeneity. We propose a new approach to community detection which we call the Spectral Clustering On Ratios-of-Eigenvectors (SCORE). Compared to classical spectral methods, the main innovation is to use the entry-wise ratios between the first leading eigenvector and each of the other leading eigenvectors for clustering. The central surprise is, the effect of degree heterogeneity is largely ancillary, and can be effectively removed by taking entry-wise ratios between the leading eigenvectors. The method is successfully applied to the web blogs data and the karate club data, with error rates of 58/1222 and 1/34, respectively. These results are much more satisfactory than those by the classical spectral methods. Also, compared to modularity methods, SCORE is computationally much faster and has smaller error rates. We develop a theoretic framework where we show that under mild conditions, the SCORE stably yields successful community detection. In the core of the analysis is the recent development on Random Matrix Theory (RMT), where the matrix-form Bernstein inequality is especially helpful."]]></description>
<dc:subject>network_data_analysis community_discovery principal_components statistics jin.jiashun to_read kith_and_kin spectral_clustering in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:2efc3caec26c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:network_data_analysis"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:community_discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:jin.jiashun"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to_read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:kith_and_kin"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:spectral_clustering"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1207.2812">
    <title>[1207.2812] Near-Optimal Algorithms for Differentially-Private Principal Components</title>
    <dc:date>2012-07-13T14:14:01+00:00</dc:date>
    <link>http://arxiv.org/abs/1207.2812</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["Principal components analysis (PCA) is a standard tool for identifying good low-dimensional approximations to data sets in high dimension. Many current data sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs between privacy and the utility of these outputs. In this paper we investigate the theory and empirical performance of differentially private approximations to PCA and propose a new method which explicitly optimizes the utility of the output. We demonstrate that on real data, there this a large performance gap between the existing methods and our method. We show that the sample complexity for the two procedures differs in the scaling with the data dimension, and that our method is nearly optimal in terms of this scaling."]]></description>
<dc:subject>to:NB principal_components dimension_reduction differential_privacy data_mining sarwate.anand</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:6a80c9e5d592/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:differential_privacy"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sarwate.anand"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1206.4628">
    <title>[1206.4628] Robust PCA in High-dimension: A Deterministic Approach</title>
    <dc:date>2012-06-23T13:51:59+00:00</dc:date>
    <link>http://arxiv.org/abs/1206.4628</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We consider principal component analysis for contaminated data-set in the high dimensional regime, where the dimensionality of each observation is comparable or even more than the number of observations. We propose a deterministic high-dimensional robust PCA algorithm which inherits all theoretical properties of its randomized counterpart, i.e., it is tractable, robust to contaminated points, easily kernelizable, asymptotic consistent and achieves maximal robustness -- a breakdown point of 50%. More importantly, the proposed method exhibits significantly better computational efficiency, which makes it suitable for large-scale real applications."]]></description>
<dc:subject>to:NB statistics principal_components high-dimensional_statistics robustness</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:354f23a07d65/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:high-dimensional_statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:robustness"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1206.4608">
    <title>[1206.4608] A Hybrid Algorithm for Convex Semidefinite Optimization</title>
    <dc:date>2012-06-23T13:45:49+00:00</dc:date>
    <link>http://arxiv.org/abs/1206.4608</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We present a hybrid algorithm for optimizing a convex, smooth function over the cone of positive semidefinite matrices. Our algorithm converges to the global optimal solution and can be used to solve general large-scale semidefinite programs and hence can be readily applied to a variety of machine learning problems. We show experimental results on three machine learning problems (matrix completion, metric learning, and sparse PCA) . Our approach outperforms state-of-the-art algorithms."]]></description>
<dc:subject>optimization machine_learning low-rank_approximation principal_components sparsity in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:14c0d9f5d1ac/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:machine_learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:low-rank_approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:in_NB"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://arxiv.org/abs/1205.7060">
    <title>[1205.7060] Sparse Principal Component Analysis with missing observations</title>
    <dc:date>2012-06-07T15:54:46+00:00</dc:date>
    <link>http://arxiv.org/abs/1205.7060</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["In this paper, we study the problem of sparse Principal Component Analysis (PCA) in the high-dimensional setting with missing observations. Our goal is to estimate the first principal component when we only have access to partial observations. Existing estimation techniques are usually derived for fully observed data sets and require a prior knowledge of the sparsity of the first principal component in order to achieve good statistical guarantees. Our contributions is threefold. First, we establish the first information-theoretic lower bound for the sparse PCA problem with missing observations. Second, we propose a simple procedure that does not require any prior knowledge on the sparsity of the unknown first principal component or any imputation of the missing observations, adapts to the unknown sparsity of the first principal component and achieves the optimal rate of estimation up to a logarithmic factor. Third, if the covariance matrix of interest admits a sparse first principal component and is in addition approximately low-rank, then we can derive a completely data-driven procedure computationally tractable in high-dimension, adaptive to the unknown sparsity of the first principal component and statistically optimal (up to a logarithmic factor)."]]></description>
<dc:subject>to:NB dimension_reduction missing_data sparsity principal_components statistics data_analysis</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:940759f8ce3e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:to:NB"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:dimension_reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:missing_data"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:sparsity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:principal_components"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:cshalizi/t:data_analysis"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://prl.aps.org/abstract/PRL/v108/i20/e200601">
    <title>Phys. Rev. Lett. 108, 200601 (2012): Number of Relevant Directions in Principal Component Analysis and Wishart Random Matrices</title>
    <dc:date>2012-05-20T20:34:49+00:00</dc:date>
    <link>http://prl.aps.org/abstract/PRL/v108/i20/e200601</link>
    <dc:creator>cshalizi</dc:creator><description><![CDATA["We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that P(N+,N)≈exp⁡[-βN2ψζ(N+/N)], where β is the Dyson index of the ensemble and ψζ(κ) is a rate function that we compute explicitly in the full range 0≤κ≤1 and for any ζ. The rate function ψζ(κ) displays a quadratic behavior modulated by a logarithmic singularity close to its minimum κ⋆(ζ). This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance Δ(N) of the number of relevant components is also shown to grow universally (independent of ζ) as Δ(N)∼(βπ2)-1ln⁡N for large N."]]></description>
<dc:subject>to_read principal_components large_deviations random_matrices stochastic_processes high-dimensional_probability re:g_paper phase_transitions in_NB</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:cshalizi/b:9f2f705f6a93/</dc:identifier>
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