Recovery of Low Rank and Jointly Sparse Matrices with Two Sampling Matrices

We provide a two-step approach to recover a jointly $k$ -sparse matrix ${\bf X}$ , (at most $k$ rows of ${\bf X}$ are nonzero), with rank $r from its under sampled measurements. Unlike the classical recovery algorithms that use the same measurement matrix for every column of ${\bf X}$ , the proposed algorithm comprises two stages, in each of which the measurement is taken by a different measurement matrix. The first stage uses a standard algorithm, to recover any $r$ columns (e.g. the first $r$ ) of ${\bf X}$ . The second uses a new set of measurements and the subspace estimate provided by these columns to recover the rest. We derive conditions on the second measurement matrix to guarantee perfect subspace aware recovery for two cases: First a worst-case setting that applies to all matrices. The second a generic case that works for almost all matrices. We demonstrate both theoretically and through simulations that when $r our approach needs far fewer measurements. It compares favorably with recent results using dense linear combinations, that do not use column-wise measurements.