Recovering Jointly Sparse Signals via Joint Basis Pursuit

This work considers recovery of signals that are sparse over two bases. For instance, a signal might be sparse in both time and frequency, or a matrix can be low rank and sparse simultaneously. To facilitate recovery, we consider minimizing the sum of the ℓ_1-norms that correspond to each basis, which is a tractable convex approach. We find novel optimality conditions which indicates a gain over traditional approaches where ℓ_1 minimization is done over only one basis. Next, we analyze these optimality conditions for the particular case of time-frequency bases. Denoting sparsity in the first and second bases by k_1,k_2 respectively, we show that, for a general class of signals, using this approach, one requires as small as O(max{k_1,k_2} log log n) measurements for successful recovery hence overcoming the classical requirement of Θ(min{k_1,k_2} log (n/(min{k_1,k_2})) for ℓ _1 minimization when k_1 ≈ k_2. Extensive simulations show that, our analysis is approximately tight