Multivariate binary data is becoming abundant in current biological research. Logistic principal component analysis (PCA) is one of the commonly used tools to explore the relationships inside a multivariate binary data set by exploiting the underlying low rank structure. We re-expressed the logistic PCA model based on the latent variable interpretation of the generalized linear model on binary data. The multivariate binary data set is assumed to be the sign observation of an unobserved quantitative data set, on which a low rank structure is assumed to exist. However, the standard logistic PCA model (using exact low rank constraint) is prone to overfitting, which could lead to divergence of some estimated parameters towards infinity. We propose to fit a logistic PCA model through non-convex singular value thresholding to alleviate the overfitting issue. An efficient Majorization-Minimization algorithm is implemented to fit the model and a missing value based cross validation (CV) procedure is introduced for the model selection. Our experiments on realistic simulations of imbalanced binary data and low signal to noise ratio show that the CV error based model selection procedure is successful in selecting the proposed model. Furthermore, the selected model demonstrates superior performance in recovering the underlying low rank structure compared to models with convex nuclear norm penalty and exact low rank constraint. A binary copy number aberration data set is used to illustrate the proposed methodology in practice.
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, an…
This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresh…
Functional binary datasets occur frequently in real practice, whereas discrete characteristics of the data can bring challenges to model estimation. In this paper, we propose a sparse logistic functional principal component analysis (SLFPCA) method to handle the functional binary data. The SLFPCA looks for local sparsity of the eigenfunctions to obtain convenience in interpretation. We formulate the problem through a penalized Bernoulli likelihood with both roughness penalty and sparseness penalty terms. An efficient algorithm is developed for the optimization of the penalized likelihood using…
Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson's formulation of a low dimensional data representation with minimum error to binary data. Our formulation does not require a matrix factorization, as previous methods do, but instead looks for projections of th…
Sequential or online dimensional reduction is of interests due to the explosion of streaming data based applications and the requirement of adaptive statistical modeling, in many emerging fields, such as the modeling of energy end-use profile. Principal Component Analysis (PCA), is the classical way of dimensional reduction. However, traditional Singular Value Decomposition (SVD) based PCA fails to model data which largely deviates from Gaussian distribution. The Bregman Divergence was recently introduced to achieve a generalized PCA framework. If the random variable under dimensional reductio…
We develop a new principal components analysis (PCA) type dimension reduction method for binary data. Different from the standard PCA which is defined on the observed data, the proposed PCA is defined on the logit transform of the success probabilities of the binary observations. Sparsity is introduced to the principal component (PC) loading vectors for enhanced interpretability and more stable extraction of the principal components. Our sparse PCA is formulated as solving an optimization problem with a criterion function motivated from a penalized Bernoulli likelihood. A Majorization--Minimiz…
In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\mathbf{v}_1,\dots,\mathbf{v}_r$ has at most $s_0$ non-zero entries. We are particularly interested in the high dimensional regime wherein $p$ is comparable to, or even much larger than $n$. In an influential paper, \cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\mathbf{v}_1,\dots,\mathbf{…
In this paper, we analyze the generalization performance of the Iterative Hard Thresholding (IHT) algorithm widely used for sparse recovery problems. The parameter estimation and sparsity recovery consistency of IHT has long been known in compressed sensing. From the perspective of statistical learning, another fundamental question is how well the IHT estimation would predict on unseen data. This paper makes progress towards answering this open question by introducing a novel sparse generalization theory for IHT under the notion of algorithmic stability. Our theory reveals that: 1) under natur…
Monitoring multichannel profiles has important applications in manufacturing systems improvement, but it is non-trivial to develop efficient statistical methods due to two main challenges. First, profiles are high-dimensional functional data with intrinsic inner- and inter-channel correlations, and one needs to develop a dimension reduction method that can deal with such intricate correlations for the purpose of effective monitoring. The second, and probably more fundamental, challenge is that the functional structure of multi-channel profiles might change over time, and thus the dimension red…
Recently, the robustification of principal component analysis has attracted lots of attention from statisticians, engineers and computer scientists. In this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation. Based on a mathematical formulation of such transformed outliers, a novel robust orthogonal complement principal component analysis (ROC-PCA) is proposed. The framework combines the popular sparsity-enforcing and low rank regularization techniques to deal with row-wise outliers a…
Functional principal component analysis (FPCA) is a fundamental tool and has attracted increasing attention in recent decades, while existing methods are restricted to data with a single or finite number of random functions (much smaller than the sample size $n$). In this work, we focus on high-dimensional functional processes where the number of random functions $p$ is comparable to, or even much larger than $n$. Such data are ubiquitous in various fields such as neuroimaging analysis, and cannot be properly modeled by existing methods. We propose a new algorithm, called sparse FPCA, which is…
The University of Tokyo（东京大学）；Kyoto University（京都大学）
We theoretically analyze the model selection consistency of least absolute shrinkage and selection operator (Lasso), both with and without post-thresholding, for high-dimensional Ising models. For random regular (RR) graphs of size $p$ with regular node degree $d$ and uniform couplings $θ_0$, it is rigorously proved that Lasso \textit{without post-thresholding} is model selection consistent in the whole paramagnetic phase with the same order of sample complexity $n=Ω{(d^3\log{p})}$ as that of $\ell_1$-regularized logistic regression ($\ell_1$-LogR). This result is consistent with the conjectur…
Estimating the leading principal components of data, assuming they are sparse, is a central task in modern high-dimensional statistics. Many algorithms were developed for this sparse PCA problem, from simple diagonal thresholding to sophisticated semidefinite programming (SDP) methods. A key theoretical question is under what conditions can such algorithms recover the sparse principal components? We study this question for a single-spike model with an $\ell_0$-sparse eigenvector, in the asymptotic regime as dimension $p$ and sample size $n$ both tend to infinity. Amini and Wainwright [Ann. Sta…
Stochastic principal component analysis (SPCA) has become a popular dimensionality reduction strategy for large, high-dimensional datasets. We derive a simplified algorithm, called Lazy SPCA, which has reduced computational complexity and is better suited for large-scale distributed computation. We prove that SPCA and Lazy SPCA find the same approximations to the principal subspace, and that the pairwise distances between samples in the lower-dimensional space is invariant to whether SPCA is executed lazily or not. Empirical studies find downstream predictive performance to be identical for bo…
In this paper we propose a principal component Liu-type logistic estimator by combining the principal component logistic regression estimator and Liu-type logistic estimator to overcome the multicollinearity problem. The superiority of the new estimator over some related estimators are studied under the asymptotic mean squared error matrix. A Monte Carlo simulation experiment is designed to compare the performances of the estimators using mean squared error criterion. Finally, a conclusion section is presented.
Biplot Methods allow for the simultaneous representation of individuals and variables of a data matrix. For Binary or Nominal data, Logistic biplots have been recently developed to extend the classical linear representations for continuous data. When data are ordinal, linear, binary or nominal logistic biplots are not adequate and techniques as Categorical Principal Component Analysis (CATPCA) or Item Response Theory (IRT) for ordinal items should be used instead. In this paper we extend the Biplot to ordinal data. The resulting method is termed Ordinal Logistic Biplot (OLB). Row scores are co…
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corru…
We proposed a general Principal Orthogonal complEment Thresholding (POET) framework for large-scale covariance matrix estimation based on an approximate factor model. A set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms were brought up to better understand how POET works. Such a framework allows us to recover the results for sub-Gaussian in a more transparent way that only depends on the concentration properties of the sample covariance matrix. As a new theoretical contribution, for the first time, such a framework all…
We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spike magnitude of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights into the roles the leading eigenvalues, sample size, and dimensionality play in principal component analysis. The results are proven by a technical device, which swaps the role of rows and columns and converts the high-dimensional problems into low-dimensional one…
The association between a person's physical activity and various health outcomes is an area of active research. The National Health and Nutrition Examination Survey (NHANES) data provide a valuable resource for studying these associations. NHANES accelerometry data has been used by many to measure individuals' activity levels. A common approach for analyzing accelerometry data is functional principal component analysis (FPCA). The first part of the paper uses Poisson FPCA (PFPCA), Gaussian FPCA (GFPCA), and nonnegative and regularized function decomposition (NARFD) to extract features from the…
Networked sensing, where the goal is to perform complex inference using a large number of inexpensive and decentralized sensors, has become an increasingly attractive research topic due to its applications in wireless sensor networks and internet-of-things. To reduce the communication, sensing and storage complexity, this paper proposes a simple sensing and estimation framework to faithfully recover the principal subspace of high-dimensional data streams using a collection of binary measurements from distributed sensors, without transmitting the whole data. The binary measurements are designed…
Sparse PCA: Phase Transitions in the Critical Regime
Stern School of Business, New York University（纽约大学斯特恩商学院）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）
Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$'' setting, in which the problem dimension $p$ is comparable to or larger than the sample size $n$. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most $k$ nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a $k…
Principal component analysis (PCA) is a well-established method commonly used to explore and visualise data. A classical PCA model is the fixed effect model where data are generated as a fixed structure of low rank corrupted by noise. Under this model, PCA does not provide the best recovery of the underlying signal in terms of mean squared error. Following the same principle as in ridge regression, we propose a regularised version of PCA that boils down to threshold the singular values. Each singular value is multiplied by a term which can be seen as the ratio of the signal variance over the t…
Brain decoding involves the determination of a subject's cognitive state or an associated stimulus from functional neuroimaging data measuring brain activity. In this setting the cognitive state is typically characterized by an element of a finite set, and the neuroimaging data comprise voluminous amounts of spatiotemporal data measuring some aspect of the neural signal. The associated statistical problem is one of classification from high-dimensional data. We explore the use of functional principal component analysis, mutual information networks, and persistent homology for examining the data…
Independent component analysis (ICA) is popular in many applications, including cognitive neuroscience and signal processing. Due to computational constraints, principal component analysis is used for dimension reduction prior to ICA (PCA+ICA), which could remove important information. The problem is that interesting independent components (ICs) could be mixed in several principal components that are discarded and then these ICs cannot be recovered. We formulate a linear non-Gaussian component model with Gaussian noise components. To estimate this model, we propose likelihood component analysi…
Sparse regularized regression methods are now widely used in genome-wide association studies (GWAS) to address the multiple testing burden that limits discovery of potentially important predictors. Linear mixed models (LMMs) have become an attractive alternative to principal components (PC) adjustment to account for population structure and relatedness in high-dimensional penalized models. However, their use in binary trait GWAS rely on the invalid assumption that the residual variance does not depend on the estimated regression coefficients. Moreover, LMMs use a single spectral decomposition …
University College London（伦敦大学学院）；Centre National de la Recherche Scientifique (CNRS)（法国国家科学研究中心）
This paper presents a method of exploring the relative predictive importance of individual variables in multicollinear data sets at three levels of significance: strong importance, moderate importance, and no importance. Implementation of Bonferroni adjustment to control for Type I error in the method is described, and results with and without the correction are compared. An example of the method in binary logistic modeling is demonstrated by using a set of 20 acoustic features to discriminate vocalic nasality in the speech of six speakers of the Mixean variety of Low Navarrese Basque. Validat…
Institute of Medical Genetics, University Medical Centre Ljubljana（卢布尔雅那大学医学中心医学遗传学研究所）
Class prediction is an important application of microarray gene expression data analysis. The high-dimensionality of microarray data, where number of genes (variables) is very large compared to the number of samples (obser- vations), makes the application of many prediction techniques (e.g., logistic regression, discriminant analysis) difficult. An efficient way to solve this prob- lem is by using dimension reduction statistical techniques. Increasingly used in psychology-related applications, Rasch model (RM) provides an appealing framework for handling high-dimensional microarray data. In th…
Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such embeddings cannot capture local structure arising in complex networks. In particular, they show that any network generated from a natural low-dimensional model cannot be both sparse and have high triangle density (high clustering coefficient), two hallmark properties of many real-world networks. In this work we show that the results of Seshadhri et al. are intimate…
We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). A theoretical analysis for both fixed design and random design settings is provided. Obt…
Technion - Israel Institute of Technology（以色列理工学院）
We consider optimization problems in which the goal is find a $k$-dimensional subspace of $\mathbb{R}^n$, $k<<n$, which minimizes a convex and smooth loss. Such problems generalize the fundamental task of principal component analysis (PCA) to include robust and sparse counterparts, and logistic PCA for binary data, among others. This problem could be approached either via nonconvex gradient methods with highly-efficient iterations, but for which arguing about fast convergence to a global minimizer is difficult or, via a convex relaxation for which arguing about convergence to a global minimize…
We propose a penalized orthogonal-components regression (POCRE) for large p small n data. Orthogonal components are sequentially constructed to maximize, upon standardization, their correlation to the response residuals. A new penalization framework, implemented via empirical Bayes thresholding, is presented to effectively identify sparse predictors of each component. POCRE is computationally efficient owing to its sequential construction of leading sparse principal components. In addition, such construction offers other properties such as grouping highly correlated predictors and allowing for…
We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + λxx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + βxx^\top)$, respectively). Prior work has shown that when the signal-to-noise ratio ($λ$ or $β\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = ρ$, it is possible to recover $x$ in polynomial time if $ρ\less…
Envelope methods offer targeted dimension reduction for various models. The overarching goal is to improve efficiency in multivariate parameter estimation by projecting the data onto a lower-dimensional subspace known as the envelope. Envelope approaches have advantages in analyzing data with highly correlated variables, but their iterative Grassmannian optimization algorithms do not scale very well with ultra high-dimensional data. While the connections between envelopes and partial least squares in multivariate linear regression have promoted recent progress in high-dimensional studies of en…
