In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. Specifically, the method achieves polynomial time complexity for certain convex cones equipped with fast projection such as the monotone cone. It attains a small error when the noisy matrix has a small cone-restricted operator norm. We supplement the above results with a minimax lower bound of the error under the spiked covariance model. Our numerical experiments on simulated and real data, show that our method achieves shorter run time and smaller error in comparison to the ordinary power iteration and some sparse principal component analysis algorithms if the principal eigenvector is in a convex cone.
Most high-dimensional matrix recovery problems are studied under the assumption that the target matrix has certain intrinsic structures. For image data related matrix recovery problems, approximate low-rankness and smoothness are the two most commonly imposed structures. For approximately low-rank matrix recovery, the robust principal component analysis (PCA) is well-studied and proved to be effective. For smooth matrix problem, 2d fused Lasso and other total variation based approaches have played a fundamental role. Although both low-rankness and smoothness are key assumptions for image data …
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 …
Random projection is widely used as a method of dimension reduction. In recent years, its combination with standard techniques of regression and classification has been explored. Here we examine its use with principal component analysis (PCA) and subspace detection methods. Specifically, we show that, under appropriate conditions, with high probability the magnitude of the residuals of a PCA analysis of randomly projected data behaves comparably to that of the residuals of a similar PCA analysis of the original data. Our results indicate the feasibility of applying subspace-based anomaly detec…
We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, e…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex factorized optimization problem. The goal of this problem is to provide a general theoretical framework for understanding when such methods work well, and to characterize the nature of the resulting fixed point. We provide a simple set of conditions under which projected gradient descent…
This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employs principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accu…
Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first $K$ principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank $K$ projections. Thus, PCA provides optimal low-rank representations of data for least-squares linear regression under standard modeling assumptions. On the other hand, when the loss function for a prediction problem is not the l…
One technique to visualize the training of neural networks is to perform PCA on the parameters over the course of training and to project to the subspace spanned by the first few PCA components. In this paper we compare this technique to the PCA of a high dimensional random walk. We compute the eigenvalues and eigenvectors of the covariance of the trajectory and prove that in the long trajectory and high dimensional limit most of the variance is in the first few PCA components, and that the projection of the trajectory onto any subspace spanned by PCA components is a Lissajous curve. We genera…
Huazhong University of Science and Technology（华中科技大学）；University of Technology Sydney（悉尼科技大学）
Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label inf…
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…
Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace chan…
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 …
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院与运筹学中心）；Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）
We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's diameter, we argue that the method performs well when initialized with a second-order-cone approximation, instead of a linear approximation. We invoke the method to provide bound gaps of 0.5-6.5% for sparse PCA problems with $1000$s of covariates, and solve nuclear norm problems over 500x500 matrices.
In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied on this purpose, it can only provide meaningful results if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments quantities and the covariance matrix can only capture the full dependence structure for multivariate Gaussian vectors. Whenever the application at hand can not satisfy t…
Purdue University（普渡大学）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
Motivated by the analysis of nonnegative data objects, a novel Nested Nonnegative Cone Analysis (NNCA) approach is proposed to overcome some drawbacks of existing methods. The application of traditional PCA/SVD method to nonnegative data often cause the approximation matrix leave the nonnegative cone, which leads to non-interpretable and sometimes nonsensical results. The nonnegative matrix factorization (NMF) approach overcomes this issue, however the NMF approximation matrices suffer several drawbacks: 1) the factorization may not be unique, 2) the resulting approximation matrix at a specifi…
Sparse PCA (SPCA) is a fundamental model in machine learning and data analytics, which has witnessed a variety of application areas such as finance, manufacturing, biology, healthcare. To select a prespecified-size principal submatrix from a covariance matrix to maximize its largest eigenvalue for the better interpretability purpose, SPCA advances the conventional PCA with both feature selection and dimensionality reduction. This paper proposes two exact mixed-integer SDPs (MISDPs) by exploiting the spectral decomposition of the covariance matrix and the properties of the largest eigenvalues. …
California Institute of Technology（加州理工学院）；University of California, Irvine（加州大学欧文分校）
High-dimensional representations often have a lower dimensional underlying structure. This is particularly the case in many decision making settings. For example, when the representation of actions is generated from a deep neural network, it is reasonable to expect a low-rank structure whereas conventional structures like sparsity are not valid anymore. Subspace recovery methods, such as Principle Component Analysis (PCA) can find the underlying low-rank structures in the feature space and reduce the complexity of the learning tasks. In this work, we propose Projected Stochastic Linear Bandit …
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 unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness conditions. Based on projected gradient descent and the double thresholding operator, our proposed generic algorithm is guaranteed to converge to the unknown low-rank and sparse matrices at a locally linear rate, while matching the best-known robustness guarantee (i.e., tolerance for sparsity). At the core of our theory is a novel structural Lipschitz gradient con…
The problem of Non-Gaussian Component Analysis (NGCA) is about finding a maximal low-dimensional subspace $E$ in $\mathbb{R}^n$ so that data points projected onto $E$ follow a non-gaussian distribution. Although this is an appropriate model for some real world data analysis problems, there has been little progress on this problem over the last decade. In this paper, we attempt to address this state of affairs in two ways. First, we give a new characterization of standard gaussian distributions in high-dimensions, which lead to effective tests for non-gaussianness. Second, we propose a simple a…
Many statistical estimation techniques for high-dimensional or functional data are based on a preliminary dimension reduction step, which consists in projecting the sample $\bX_1, \hdots, \bX_n$ onto the first $D$ eigenvectors of the Principal Component Analysis (PCA) associated with the empirical projector $\hat Π_D$. Classical nonparametric inference methods such as kernel density estimation or kernel regression analysis are then performed in the (usually small) $D$-dimensional space. However, the mathematical analysis of this data-driven dimension reduction scheme raises technical problems,…
The high-dimensional data setting, in which p >> n, is a challenging statistical paradigm that appears in many real-world problems. In this setting, learning a compact, low-dimensional representation of the data can substantially help distinguish signal from noise. One way to achieve this goal is to perform subspace learning to estimate a small set of latent features that capture the majority of the variance in the original data. Most existing subspace learning models, such as PCA, assume that the data can be fully represented by its embedding in one or more latent subspaces. However, in this …
