The robust principal component analysis (RPCA) decomposes a data matrix into a low-rank part and a sparse part. There are mainly two types of algorithms for RPCA. The first type of algorithm applies regularization terms on the singular values of a matrix to obtain a low-rank matrix. However, calculating singular values can be very expensive for large matrices. The second type of algorithm replaces the low-rank matrix as the multiplication of two small matrices. They are faster than the first type because no singular value decomposition (SVD) is required. However, the rank of the low-rank matrix is required, and an accurate rank estimation is needed to obtain a reasonable solution. In this paper, we propose algorithms that combine both types. Our proposed algorithms require an upper bound of the rank and SVD on small matrices. First, they are faster than the first type because the cost of SVD on small matrices is negligible. Second, they are more robust than the second type because an upper bound of the rank instead of the exact rank is required. Furthermore, we apply the Gauss-Newton method to increase the speed of our algorithms. Numerical experiments show the better performance of our proposed algorithms.
University of California, Los Angeles（加州大学洛杉矶分校）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Sun Yat-sen University（中山大学）
Robust principal component analysis (RPCA) is a widely used tool for dimension reduction. In this work, we propose a novel non-convex algorithm, coined Iterated Robust CUR (IRCUR), for solving RPCA problems, which dramatically improves the computational efficiency in comparison with the existing algorithms. IRCUR achieves this acceleration by employing CUR decomposition when updating the low rank component, which allows us to obtain an accurate low rank approximation via only three small submatrices. Consequently, IRCUR is able to process only the small submatrices and avoid expensive computin…
The University of California, Davis（加州大学戴维斯分校）；Google AI（谷歌人工智能）；Princeton University（普林斯顿大学）
Sparse PCA is a widely used technique for high-dimensional data analysis. In this paper, we propose a new method called low-rank principal eigenmatrix analysis. Different from sparse PCA, the dominant eigenvectors are allowed to be dense but are assumed to have a low-rank structure when matricized appropriately. Such a structure arises naturally in several practical cases: Indeed the top eigenvector of a circulant matrix, when matricized appropriately is a rank-1 matrix. We propose a matricized rank-truncated power method that could be efficiently implemented and establish its computational an…
In 2018, at the World Economic Forum in Davos it was presented a new countries' economic performance metric named the Inclusive Development Index (IDI) composed of 12 indicators. The new metric implies that countries might need to realize structural reforms for improving both economic expansion and social inclusion performance. That is why, it is vital for the IDI calculation method to have strong statistical and mathematical basis, so that results are accurate and transparent for public purposes. In the current work, we propose a novel approach for the IDI estimation - the Ranking Relative Pr…
A few iterations of alternating least squares with a random starting point provably suffice to produce nearly optimal spectral- and Frobenius-norm accuracies of low-rank approximations to a matrix; iterating to convergence is unnecessary. Thus, software implementing alternating least squares can be retrofitted via appropriate setting of parameters to calculate nearly optimally accurate low-rank approximations highly efficiently, with no need for convergence.
University of California, Berkeley（加州大学伯克利分校）；Tsinghua-Berkeley Shenzhen Institute（清华-伯克利深圳学院）
This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the RPCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the RPCA as a non-convex and non-smooth $…
We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
Principal components analysis (PCA) is a widely used dimension reduction technique with an extensive range of applications. In this paper, an online distributed algorithm is proposed for recovering the principal eigenspaces. We further establish its rate of convergence and show how it relates to the number of nodes employed in the distributed computation, the effective rank of the data matrix under consideration, and the gap in the spectrum of the underlying population covariance matrix. The proposed algorithm is illustrated on low-rank approximation and $\boldsymbol{k}$-means clustering tasks…
Matrix completion constantly receives tremendous attention from many research fields. It is commonly applied for recommender systems such as movie ratings, computer vision such as image reconstruction or completion, multi-task learning such as collaboratively modeling time-series trends of multiple sensors, and many other applications. Matrix completion techniques are usually computationally exhaustive and/or fail to capture the heterogeneity in the data. For example, images usually contain a heterogeneous set of objects, and thus it is a challenging task to reconstruct images with high levels…
Tensor completion and robust principal component analysis have been widely used in machine learning while the key problem relies on the minimization of a tensor rank that is very challenging. A common way to tackle this difficulty is to approximate the tensor rank with the $\ell_1-$norm of singular values based on its Tensor Singular Value Decomposition (T-SVD). Besides, the sparsity of a tensor is also measured by its $\ell_1-$norm. However, the $\ell_1$ penalty is essentially biased and thus the result will deviate. In order to sidestep the bias, we propose a novel non-convex tensor rank sur…
Robust principal component analysis (RPCA) can recover low-rank matrices when they are corrupted by sparse noises. In practice, many matrices are, however, of high-rank and hence cannot be recovered by RPCA. We propose a novel method called robust kernel principal component analysis (RKPCA) to decompose a partially corrupted matrix as a sparse matrix plus a high or full-rank matrix with low latent dimensionality. RKPCA can be applied to many problems such as noise removal and subspace clustering and is still the only unsupervised nonlinear method robust to sparse noises. Our theoretical analys…
This paper considers sparse spiked covariance matrix models in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the pri…
We consider the problem of estimating multiple principal components using the recently-proposed Sparse and Functional Principal Components Analysis (SFPCA) estimator. We first propose an extension of SFPCA which estimates several principal components simultaneously using manifold optimization techniques to enforce orthogonality constraints. While effective, this approach is computationally burdensome so we also consider iterative deflation approaches which take advantage of existing fast algorithms for rank-one SFPCA. We show that alternative deflation schemes can more efficiently extract sign…
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 …
Various problems in data analysis and statistical genetics call for recovery of a column-sparse, low-rank matrix from noisy observations. We propose ReFACTor, a simple variation of the classical Truncated Singular Value Decomposition (TSVD) algorithm. In contrast to previous sparse principal component analysis (PCA) algorithms, our algorithm can provably reveal a low-rank signal matrix better, and often significantly better, than the widely used TSVD, making it the algorithm of choice whenever column-sparsity is suspected. Empirically, we observe that ReFACTor consistently outperforms TSVD eve…
Carnegie Mellon University（卡内基梅隆大学）；National University of Singapore（新加坡国立大学）；Cornell University（康奈尔大学）；Tencent AI Lab（腾讯人工智能实验室）；Peking University（北京大学）
In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix ca…
Carnegie Mellon University（卡内基梅隆大学）；National University of Singapore（新加坡国立大学）
This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank and the corresponding tensor nuclear norm. Then we solve the TRPCA problem by convex optimization whose objective is a weighted combination of the new tensor nuclear norm and the $\ell_1$-norm. In theory, we show that under certain incoherence conditions, the convex program exac…
High-dimensional multi-source data are encountered in many fields. Despite recent developments on the integrative dimension reduction of such data, most existing methods cannot easily accommodate data of multiple types (e.g., binary or count-valued). Moreover, multi-source data often have block-wise missing structure, i.e., data in one or more sources may be completely unobserved for a sample. The heterogeneous data types and presence of block-wise missing data pose significant challenges to the integration of multi-source data and further statistical analyses. In this paper, we develop a low-…
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…
We propose a nested reduced-rank regression (NRRR) approach in fitting regression model with multivariate functional responses and predictors, to achieve tailored dimension reduction and facilitate interpretation/visualization of the resulting functional model. Our approach is based on a two-level low-rank structure imposed on the functional regression surfaces. A global low-rank structure identifies a small set of latent principal functional responses and predictors that drives the underlying regression association. A local low-rank structure then controls the complexity and smoothness of the…
In the era of big data, reducing data dimensionality is critical in many areas of science. Widely used Principal Component Analysis (PCA) addresses this problem by computing a low dimensional data embedding that maximally explain variance of the data. However, PCA has two major weaknesses. Firstly, it only considers linear correlations among variables (features), and secondly it is not suitable for categorical data. We resolve these issues by proposing Maximally Correlated Principal Component Analysis (MCPCA). MCPCA computes transformations of variables whose covariance matrix has the largest …
University of Michigan（密歇根大学）；University of California, Davis（加州大学戴维斯分校）
Singular value decomposition (SVD) based principal component analysis (PCA) breaks down in the high-dimensional and limited sample size regime below a certain critical eigen-SNR that depends on the dimensionality of the system and the number of samples. Below this critical eigen-SNR, the estimates returned by the SVD are asymptotically uncorrelated with the latent principal components. We consider a setting where the left singular vector of the underlying rank one signal matrix is assumed to be sparse and the right singular vector is assumed to be equisigned, that is, having either only nonneg…
A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?
Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen（拉德堡德大学尼梅根大脑、认知与行为多德斯研究所）；Department of Mathematics, FEW, VU university Amsterdam（阿姆斯特丹自由大学FEW数学系）；Unit for BMI, North-West University（西北大学BMI单元）
University Politehnica of Bucharest（布加勒斯特理工大学）；University of Bucharest（布加勒斯特大学）
In this paper, we describe a new algorithm to build a few sparse principal components from a given data matrix. Our approach does not explicitly create the covariance matrix of the data and can be viewed as an extension of the Kogbetliantz algorithm to build an approximate singular value decomposition for a few principal components. We show the performance of the proposed algorithm to recover sparse principal components on various datasets from the literature and perform dimensionality reduction for classification applications.
Strongly Convex Programming for Principal Component Pursuit
the School of Electronic Engineering University of Electronic Science and Technology of China（电子科技大学电子工程学院）；Department of Electrical Engineering, ESAT-SCD / IBBT - KU Leuven Future Health Department, KU Leuven（荷语鲁汶大学电气工程系、ESAT-SCD/IBBT未来健康部门）
Variational Bayesian Inference for Tensor Robust Principal Component Analysis
Department of Statistics and Data Science, Southern University of Science and Technology（统计与数据科学系，南方科技大学）；Lingnan University（岭南大学）；Key Laboratory of Computing and Stochastic Mathematics (LCSM), School of Mathematics and Statistics, Hunan Normal University（计算与随机数学重点实验室（LCSM），数学与统计学学院，湖南师范大学）
Accelerated Alternating Projections for Robust Principal Component Analysis
Department of Mathematics, University of California, Los Angeles（加州大学洛杉矶分校数学系）；Department of Mathematics, Hong Kong University of Science and Technology（香港理工大学数学系）；School of Data Science, Fudan University（复旦大学数据科学学院）
