A Nonconvex Projection Method for Robust PCA

Robust principal component analysis (RPCA) is a well-studied problem with the goal of decomposing a matrix into the sum of low-rank and sparse components. In this paper, we propose a nonconvex feasibility reformulation of RPCA problem and apply an alternating projection method to solve it. To the best of our knowledge, we are the first to propose a method that solves RPCA problem without considering any objective function, convex relaxation, or surrogate convex constraints. We demonstrate through extensive numerical experiments on a variety of applications, including shadow removal, background estimation, face detection, and galaxy evolution, that our approach matches and often significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19

Matrix Completion with Noise via Leveraged Sampling

Many matrix completion methods assume that the data follows the uniform distribution. To address the limitation of this assumption, Chen et al. \cite{Chen20152999} propose to recover the matrix where the data follows the specific biased distribution. Unfortunately, in most real-world applications, the recovery of a data matrix appears to be incomplete, and perhaps even corrupted information. This paper considers the recovery of a low-rank matrix, where some observed entries are sampled in a \emph{biased distribution} suitably dependent on \emph{leverage scores} of a matrix, and some observed entries are uniformly corrupted. Our theoretical findings show that we can provably recover an unknown $n\times n$ matrix of rank $r$ from just about $O(nr\log^2 n)$ entries even when the few observed entries are corrupted with a small amount of noisy information. Empirical studies verify our theoretical results

Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis

Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.Comment: To appear in IEEE Transactions on Information Theor