Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach

We study the Sparse Plus Low Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix $\mathbf{D}$ into a sparse matrix $\mathbf{Y}$ containing the perturbations plus a low rank matrix $\mathbf{X}$. SLR is a fundamental problem in Operations Research and Machine Learning arising in many applications such as data compression, latent semantic indexing, collaborative filtering and medical imaging. We introduce a novel formulation for SLR that directly models the underlying discreteness of the problem. For this formulation, we develop an alternating minimization heuristic to compute high quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We further develop a custom branch and bound routine that leverages our heuristic and convex relaxation that solves small instances of SLR to certifiable near-optimality. Our heuristic can scale to $n=10000$ in hours, our relaxation can scale to $n=200$ in hours, and our branch and bound algorithm can scale to $n=25$ in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of the MSE of the low rank matrix and that of the sparse matrix.