High Dimensional Low Rank plus Sparse Matrix Decomposition

This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data applications. Most of the existing decomposition algorithms are not applicable in high dimensional settings for two main reasons. First, they need the whole data to extract the low-rank/sparse components; second, they are based on an optimization problem whose dimensionality is equal to the dimension of the given data. In this paper, we present a randomized decomposition algorithm which exploits the low dimensional geometry of the low rank matrix to reduce the complexity. The low rank plus sparse matrix decomposition problem is reformulated as a columns-rows subspace learning problem. It is shown that when the columns/rows subspace of the low rank matrix is incoherent with the standard basis, the columns/rows subspace can be learned from a small random subset of the columns/rows of the given data matrix. Thus, the high dimensional decomposition problem is converted to a subspace learning problem (which is a low dimensional optimization problem) and it uses a small random subset of the data rather than the whole big data matrix. We derive sufficient conditions, which are no more stringent than those for existing methods, to ensure exact decomposition with high probability.