Superposed Atomic Representation for Robust High-Dimensional Data Recovery of Multiple Low-Dimensional Structures

Abstract This paper proposes a unified Superposed Atomic Representation (SAR) framework for high-dimensional data recovery with multiple low-dimensional structures. The data can be in various forms ranging from vectors to tensors. The goal of SAR is to recover different components from their sum, where each component has a low-dimensional structure, such as sparsity, low-rankness or be lying a low-dimensional subspace. Examples of SAR include, but not limited to, Robust Sparse Representation (RSR), Robust Principal Component Analysis (RPCA), Tensor RPCA (TRPCA), and Outlier Pursuit (OP). We establish the theoretical guarantee for SAR. To further improve SAR, we also develop a Weighted SAR (WSAR) framework by paying more attention and penalizing less on significant atoms of each component. An effective optimization algorithm is devised for WSAR and the convergence of the algorithm is rigorously proved. By leveraging WSAR as a general platform, several new methods are proposed for high-dimensional data recovery. The experiments on real data demonstrate the superiority of WSAR for various data recovery problems.