Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. Thispaper considers the idealized "robust principal component analysis" problem of recovering a low rank matrix A from corrupted observations D A E. Here, the corrupted entries E are unknown and the errors can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns bysolving a simple convex program, for which we give a fast and provably convergent algorithm. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation spaceand the number of errors E grows in proportion to the total number of entries in the matrix. A byproduct of our analysis is the first proportional growth results for the related problem of completing a low-rank matrix from a small fraction ofits entries. Simulations and real-data examples corroborate the theoretical results, and suggest potential applications in computer vision.

We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This \emph{structured sparse PCA} is based on a structured regularization recently introduced by [1]. While classical sparse priors only deal with \textit{cardinality}, the regularization we use encodes higher-order information about the data. We propose an efficient and simple optimization procedure to solve this problem. Experiments with two practical tasks, face recognition and the study of the dynamics of a protein complex, demonstrate the benefits of the proposed structured approach over unstructured approaches.

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in literature [15, 6, 17, 28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic, random, and real data, respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors.

In this paper, we study the application of sparse principal component analysis (PCA) to clustering and feature selection problems. Sparse PCA seeks sparse factors, or linear combinations of the data variables, explaining a maximum amount of variance in the data while having only a limited number of nonzero coefficients. PCA is often used as a simple clustering technique and sparse factors allow us here to interpret the clusters in terms of a reduced set of variables. We begin with a brief introduction and motivation on sparse PCA and detail our implementation of the algorithm in d'Aspremont et al. (2005). We then apply these results to some classic clustering and feature selection problems arising in biology.

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. We prove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence this allows to infer stability results for these estimated spaces.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. We prove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence this allows to infer stability results for these estimated spaces.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. Weprove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence thisallows to infer stability results for these estimated spaces.

Eigenvoice speaker adaptation has been shown to be effective when only a small amount of adaptation data is available. At the heart of the method is principal component analysis (PCA) employed to find the most important eigenvoices. In this paper, we postulate that nonlinear PCA, in particular kernel PCA, may be even more effective. One major challenge is to map the feature-space eigenvoices back to the observation space so that the state observation likelihoods can be computed during the estimation of eigenvoice weights and subsequent decoding. Our solution is to compute kernel PCA using composite kernels, and we will call our new method kernel eigenvoice speaker adaptation. On the TIDIGITS corpus, we found that compared with a speaker-independent model, our kernel eigenvoice adaptation method can reduce the word error rate by 28-33% while the standard eigenvoice approach can only match the performance of the speaker-independent model.

Eigenvoice speaker adaptation has been shown to be effective when only a small amount of adaptation data is available. At the heart of the method is principal component analysis (PCA) employed to find the most important eigenvoices. In this paper, we postulate that nonlinear PCA, in particular kernel PCA, may be even more effective. One major challenge is to map the feature-space eigenvoices back to the observation space so that the state observation likelihoods can be computed during the estimation of eigenvoice weights and subsequent decoding. Our solution is to compute kernel PCA using composite kernels, and we will call our new method kernel eigenvoice speaker adaptation. On the TIDIGITS corpus, we found that compared with a speaker-independent model, our kernel eigenvoice adaptation method can reduce the word error rate by 28-33% while the standard eigenvoice approach can only match the performance of the speaker-independent model.