A-T-LAS$_{2,1}$: A Multi-Penalty Approach to Compressed Sensing of Low-Rank Matrices with Sparse Decompositions

Compressed Sensing is about recovering an unknown vector of dimension $n$ from $m \ll n$ linear measurements. This task becomes possible, for instance, when few entries of the vector have large magnitude and, hence, the vector is essentially of low intrinsic dimension. If one wishes to recover an $n_1\times n_2$ matrix instead, low-rankness can be added as sparsity structure of low intrinsic dimensionality. For this purpose we propose a novel algorithm, which we call Alternating Tikhonov regularization and Lasso (A-T-LAS$_{2,1}$). It is based on a multi-penalty regularization for recovery of low-rank matrices with approximately sparse singular vectors, which is able to leverage both structures (low-rankness and sparsity) simultaneously and effectively reduce further the number of necessary measurements with respect to the sole use of one of the two sparsity structures. We provide nearly-optimal recovery guarantees of A-T-LAS$_{2,1}$. The analysis is surprisingly relatively simple, e.g., compared to the one of other similar approaches such as Sparse Power Factorization (SPF). It relies on an adaptation of restricted isometry property to low-rank and approximately sparse matrices, LASSO technique, and results on proximal alternating minimization. We show that A-T-LAS$_{2,1}$ is more efficient than convex relaxation and exhibits similar performance to the state of the art method SPF, outperforming it in strong noise regimes and for matrices whose singular vectors do not possess exact (joint-) sparse support. Moreover, contrary to SPF, A-T-LAS$_{2,1}$, if properly initialized, is shown to converge even for measurements not fulfilling the restricted isometry property to matrices with some guaranteed sparsity and minimal discrepancy to data.