Dynamic inner principal component analysis (DiPCA) is a powerful method for the analysis of time-dependent multivariate data. DiPCA extracts dynamic latent variables that capture the most dominant temporal trends by solving a large-scale, dense, and nonconvex nonlinear program (NLP). A scalable decomposition algorithm has been recently proposed in the literature to solve these challenging NLPs. The decomposition algorithm performs well in practice but its convergence properties are not well understood. In this work, we show that this algorithm is a specialized variant of a coordinate maximization algorithm. This observation allows us to explain why the decomposition algorithm might work (or not) in practice and can guide improvements. We compare the performance of the decomposition strategies with that of the off-the-shelf solver Ipopt. The results show that decomposition is more scalable and, surprisingly, delivers higher quality solutions.
We consider a situation in which we see samples in $\mathbb{R}^d$ drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both.
We study the convergence properties of the VR-PCA algorithm introduced by \cite{shamir2015stochastic} for fast computation of leading singular vectors. We prove several new results, including a formal analysis of a block version of the algorithm, and convergence from random initialization. We also make a few observations of independent interest, such as how pre-initializing with just a single exact power iteration can significantly improve the runtime of stochastic methods, and what are the convexity and non-convexity properties of the underlying optimization problem.
Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
Department of Statistics University of Illinois Urbana-Champaign（统计学系伊利诺伊大学厄巴纳-香槟分校）；Department of ACMS University of Notre Dame（ACMS系诺丁汉大学）；Department of Mathematics University of British Columbia（数学系不列颠哥伦比亚大学）；Department of Mathematics Thomas Lord Department of Computer Science University of Southern California（数学系托马斯·劳德计算机科学系南加州大学）
We present Matrix Krasulina, an algorithm for online k-PCA, by generalizing the classic Krasulina's method (Krasulina, 1969) from vector to matrix case. We show, both theoretically and empirically, that the algorithm naturally adapts to data low-rankness and converges exponentially fast to the ground-truth principal subspace. Notably, our result suggests that despite various recent efforts to accelerate the convergence of stochastic-gradient based methods by adding a O(n)-time variance reduction step, for the k-PCA problem, a truly online SGD variant suffices to achieve exponential convergence…
In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the $k$-dimensional subspace which can explain the most variation of the $d$-dimensional data points that come into memory sequentially. In order to deal with large $d$ and large $N$ (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the informat…
We present an algorithm for L1-norm kernel PCA and provide a convergence analysis for it. While an optimal solution of L2-norm kernel PCA can be obtained through matrix decomposition, finding that of L1-norm kernel PCA is not trivial due to its non-convexity and non-smoothness. We provide a novel reformulation through which an equivalent, geometrically interpretable problem is obtained. Based on the geometric interpretation of the reformulated problem, we present a fixed-point type algorithm that iteratively computes a binary weight for each observation. As the algorithm requires only inner pr…
The Chinese University of Hong Kong（香港中文大学）；University of California, Davis（加利福尼亚大学戴维斯分校）；The Pennsylvania State University（宾夕法尼亚州立大学）；University of Minnesota（明尼苏达大学）
Sparse principal component analysis (PCA) and sparse canonical correlation analysis (CCA) are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as an optimization problem with nonsmooth objective and nonconvex constraints. Since non-smoothness and nonconvexity bring numerical difficulties, most algorithms suggested in the literature either solve some relaxations or are heuristic and lack convergence guarantees. In this paper, we propose a new alternating manifold proximal gradient method to solve these…
Kernel method is a very powerful tool in machine learning. The trick of kernel has been effectively and extensively applied in many areas of machine learning, such as support vector machine (SVM) and kernel principal component analysis (kernel PCA). Kernel trick is to define a kernel function which relies on the inner-product of data in the feature space without knowing these feature space data. In this paper, the kernel trick will be employed to extend the algorithm of spectrum sensing with leading eigenvector under the framework of PCA to a higher dimensional feature space. Namely, the leadi…
In this paper, we propose an acceleration scheme for online memory-limited PCA methods. Our scheme converges to the first $k>1$ eigenvectors in a single data pass. We provide empirical convergence results of our scheme based on the spiked covariance model. Our scheme does not require any predefined parameters such as the eigengap and hence is well facilitated for streaming data scenarios. Furthermore, we apply our scheme to challenging time-varying systems where online PCA methods fail to converge. Specifically, we discuss a family of time-varying systems that are based on Molecular Dynamics s…
We study the problem of recovering the subspace spanned by the first $k$ principal components of $d$-dimensional data under the streaming setting, with a memory bound of $O(kd)$. Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheless, the convergence rate of the family can be seriously affected by the learning rate of the descent steps and deserves more serious study. The second family is based on the power method over blocks of data, but setting the block size for its existing algorithms is not an easy tas…
University of Wisconsin Madison（威斯康星大学麦迪逊分校）；Amazon Lab126（亚马逊Lab126）
We present an efficient stochastic algorithm (RSG+) for canonical correlation analysis (CCA) using a reparametrization of the projection matrices. We show how this reparametrization (into structured matrices), simple in hindsight, directly presents an opportunity to repurpose/adjust mature techniques for numerical optimization on Riemannian manifolds. Our developments nicely complement existing methods for this problem which either require $O(d^3)$ time complexity per iteration with $O(\frac{1}{\sqrt{t}})$ convergence rate (where $d$ is the dimensionality) or only extract the top $1$ component…
Oja's rule [Oja, Journal of mathematical biology 1982] is a well-known biologically-plausible algorithm using a Hebbian-type synaptic update rule to solve streaming principal component analysis (PCA). Computational neuroscientists have known that this biological version of Oja's rule converges to the top eigenvector of the covariance matrix of the input in the limit. However, prior to this work, it was open to prove any convergence rate guarantee. In this work, we give the first convergence rate analysis for the biological version of Oja's rule in solving streaming PCA. Moreover, our convergen…
Microsoft Corporation（微软公司）；Conduent Labs India（Conduent印度实验室）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of California Irvine（加利福尼亚大学尔湾分校）；Xerox Research Centre India（施乐印度研究中心）
The robust PCA problem, wherein, given an input data matrix that is the superposition of a low-rank matrix and a sparse matrix, we aim to separate out the low-rank and sparse components, is a well-studied problem in machine learning. One natural question that arises is that, as in the inductive setting, if features are provided as input as well, can we hope to do better? Answering this in the affirmative, the main goal of this paper is to study the robust PCA problem while incorporating feature information. In contrast to previous works in which recovery guarantees are based on the convex rela…
Oregon State University（俄勒冈州立大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Washington State University（华盛顿州立大学）
Reinforcement learning algorithms need to deal with the exponential growth of states and actions when exploring optimal control in high-dimensional spaces. This is known as the curse of dimensionality. By projecting the agent's state onto a low-dimensional manifold, we can represent the state space in a smaller and more efficient representation. By using this representation during learning, the agent can converge to a good policy much faster. We test this approach in the Mario Benchmarking Domain. When using dimensionality reduction in Mario, learning converges much faster to a good policy. Bu…
Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
Principal Component Analysis (PCA) is a popular method for dimension reduction and has attracted an unfailing interest for decades. More recently, kernel PCA (KPCA) has emerged as an extension of PCA but, despite its use in practice, a sound theoretical understanding of KPCA is missing. We contribute several lower and upper bounds on the efficiency of KPCA, involving the empirical eigenvalues of the kernel Gram matrix and new quantities involving a notion of variance. These bounds show how much information is captured by KPCA on average and contribute a better theoretical understanding of its …
The classical non-greedy algorithm (NGA) and the recently proposed proximal alternating minimization method with extrapolation (PAMe) for $L_1$-norm PCA are revisited and their finite-step convergence are studied. It is first shown that NGA can be interpreted as a conditional subgradient or an alternating maximization method. By recognizing it as a conditional subgradient, we prove that the iterative points generated by the algorithm will be constant in finitely many steps under a certain full-rank assumption; such an assumption can be removed when the projection dimension is one. By treating …
Principal component analysis (PCA) is often used to reduce the dimension of data by selecting a few orthonormal vectors that explain most of the variance structure of the data. L1 PCA uses the L1 norm to measure error, whereas the conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the fitting error of the reconstructed data, we propose an exact reweighted and an approximate algorithm based on iteratively reweighted least squares. We provide convergence analyses, and compare their performance against benchmark algorithms in the literature. The computational experiment shows th…
We study sparse principal components analysis in the high-dimensional setting, where $p$ (the number of variables) can be much larger than $n$ (the number of observations). We prove optimal, non-asymptotic lower and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an $\ell_q$ ball for $q \in [0,1]$. Our bounds are sharp in $p$ and $n$ for all $q \in [0, 1]$ over a wide class of distributions. The upper bound is obtained by analyzing the performance of $\ell_q$-constrained PCA. In particular, our results provide convergence rates for $\ell_1$-constrain…
Harbin Institute of Technology（哈尔滨工业大学）；Qufu Normal University（曲阜师范大学）；Anhui University（安徽大学）
Principal Component Analysis (PCA) has been used to study the pathogenesis of diseases. To enhance the interpretability of classical PCA, various improved PCA methods have been proposed to date. Among these, a typical method is the so-called sparse PCA, which focuses on seeking sparse loadings. However, the performance of these methods is still far from satisfactory due to their limitation of using unsupervised learning methods; moreover, the class ambiguity within the sample is high. To overcome this problem, this study developed a new PCA method, which is named the Supervised Discriminative …
Georgia Institute of Technology（佐治亚理工学院）；Lund University（隆德大学）
We propose a new analytical approximation to the $χ^2$ kernel that converges geometrically. The analytical approximation is derived with elementary methods and adapts to the input distribution for optimal convergence rate. Experiments show the new approximation leads to improved performance in image classification and semantic segmentation tasks using a random Fourier feature approximation of the $\exp-χ^2$ kernel. Besides, out-of-core principal component analysis (PCA) methods are introduced to reduce the dimensionality of the approximation and achieve better performance at the expense of onl…
A new framework for many multiblock component methods (including consensus and hierarchical PCA) is proposed. It is based on the consensus PCA model: a scheme connecting each block of variables to a superblock obtained by concatenation of all blocks. Regularized consensus PCA is obtained by applying regularized generalized canonical correlation analysis to this scheme for the function $g(x) = x^m$ where $m \ge 1$. A gradient algorithm is proposed. At convergence, a solution of the stationary equation related to the optimization problem is obtained. For m = 1, 2 or 4 and shrinkage constants equ…
We address the problem of defining a group sparse formulation for Principal Components Analysis (PCA) - or its equivalent formulations as Low Rank approximation or Dictionary Learning problems - which achieves a compromise between maximizing the variance explained by the components and promoting sparsity of the loadings. So we propose first a new definition of the variance explained by non necessarily orthogonal components, which is optimal in some aspect and compatible with the principal components situation. Then we use a specific regularization of this variance by the group-$\ell_{1}$ norm …
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）
We study the robust principal component analysis (RPCA) problem in a distributed setting. The goal of RPCA is to find an underlying low-rank estimation for a raw data matrix when the data matrix is subject to the corruption of gross sparse errors. Previous studies have developed RPCA algorithms that provide stable solutions with fast convergence. However, these algorithms are typically hard to scale and cannot be implemented distributedly, due to the use of either SVD or large matrix multiplication. In this paper, we propose the first distributed robust principal analysis algorithm based on co…
Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric en…
University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Google Brain（谷歌大脑）
We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly Q…
Many applications, such as photon-limited imaging and genomics, involve large datasets with noisy entries from exponential family distributions. It is of interest to estimate the covariance structure and principal components of the noiseless distribution. Principal Component Analysis (PCA), the standard method for this setting, can be inefficient when the noise is non-Gaussian. We develop $e$PCA (exponential family PCA), a new methodology for PCA on exponential family distributions. $e$PCA can be used for dimensionality reduction and denoising of large data matrices. $e$PCA involves the eigend…
Phylogenetic PCA (p-PCA) is a version of PCA for observations that are leaf nodes of a phylogenetic tree. P-PCA accounts for the fact that such observations are not independent, due to shared evolutionary history. The method works on Euclidean data, but in evolutionary biology there is a need for applying it to data on manifolds, particularly shapes. We provide a generalization of p-PCA to data lying on Riemannian manifolds, called Tangent p-PCA. Tangent p-PCA thus makes it possible to perform dimension reduction on a data set of shapes, taking into account both the non-linear structure of the…
Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction and uncorrelated feature learning. Furthermore, the enormity of the dimensions and sample size in the modern day datasets have rendered the centralized PCA solutions unusable. In that vein, this paper reconsiders the problem of PCA when data samples are distributed across nodes in an arbitrarily connected network. While a few solutions for distributed PCA exi…
University of California, Irvine（加利福尼亚大学欧文分校）；Duke University（杜克大学）；Google Research（谷歌研究院）
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA…
This paper presents new algorithms to solve the feature-sparsity constrained PCA problem (FSPCA), which performs feature selection and PCA simultaneously. Existing optimization methods for FSPCA require data distribution assumptions and are lack of global convergence guarantee. Though the general FSPCA problem is NP-hard, we show that, for a low-rank covariance, FSPCA can be solved globally (Algorithm 1). Then, we propose another strategy (Algorithm 2) to solve FSPCA for the general covariance by iteratively building a carefully designed proxy. We prove theoretical guarantees on approximation …
University of California, Davis（加利福尼亚大学戴维斯分校）；Penn State（宾夕法尼亚州立大学）
Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimization problems with exploitable special structures. Many specialized efficient optimization methods have been proposed to solve robust PCA and related problems. In this paper we review existing optimization methods for solving convex and nonconvex relaxations/variants of robust PCA,…
University of Tsukuba（筑波大学）；The Institute of Statistical Mathematics（统计数理研究所）
Principal component analysis (PCA) is a widely used method for data processing, such as for dimension reduction and visualization. Standard PCA is known to be sensitive to outliers, and thus, various robust PCA methods have been proposed. It has been shown that the robustness of many statistical methods can be improved using mode estimation instead of mean estimation, because mode estimation is not significantly affected by the presence of outliers. Thus, this study proposes a modal principal component analysis (MPCA), which is a robust PCA method based on mode estimation. The proposed method …
We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-p…
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…
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…
Principal Component Analysis (PCA) is a dimension reduction technique. It produces inconsistent estimators when the dimensionality is moderate to high, which is often the problem in modern large-scale applications where algorithm scalability and model interpretability are difficult to achieve, not to mention the prevalence of missing values. While existing sparse PCA methods alleviate inconsistency, they are constrained to the Gaussian assumption of classical PCA and fail to address algorithm scalability issues. We generalize sparse PCA to the broad exponential family distributions under high-…
University of Michigan（密歇根大学）；Carnegie Mellon University（卡内基梅隆大学）；Harvard University（哈佛大学）
For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many infer…
Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of t…
Sparse principal component analysis (PCA) is an important technique for dimensionality reduction of high-dimensional data. However, most existing sparse PCA algorithms are based on non-convex optimization, which provide little guarantee on the global convergence. Sparse PCA algorithms based on a convex formulation, for example the Fantope projection and selection (FPS), overcome this difficulty, but are computationally expensive. In this work we study sparse PCA based on the convex FPS formulation, and propose a new algorithm that is computationally efficient and applicable to large and high-d…
Principal Component Analysis (PCA) is one of the most important unsupervised methods to handle high-dimensional data. However, due to the high computational complexity of its eigen decomposition solution, it hard to apply PCA to the large-scale data with high dimensionality. Meanwhile, the squared L2-norm based objective makes it sensitive to data outliers. In recent research, the L1-norm maximization based PCA method was proposed for efficient computation and being robust to outliers. However, this work used a greedy strategy to solve the eigen vectors. Moreover, the L1-norm maximization base…
In this paper, we study the power iteration algorithm for the spiked tensor model, as introduced in [44]. We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm converges, for the rank one spiked tensor model, we show the estimators for the spike strength and linear functionals of the signal are asymptotically Gaussian; for the multi-rank spiked tensor model, we show the estimators are asymptotically mixtures of Gaussian. This new phenomenon is different from the spiked matrix model. Using these asymptotic results of…
Halıcıoğlu Data Science Institute, UCSD（加州大学圣迭戈分校哈利乔格鲁数据科学学院）；Computer Science and Engineering Department, UCSD（加州大学圣迭戈分校计算机科学与工程系）；Electrical and Computer Engineering Department, UCSD（加州大学圣迭戈分校电气与计算机工程系）
