An Information-Theoretic Study for Joint Sparsity Pattern Recovery With Different Sensing Matrices

In this paper, we study a support set reconstruction problem for multiple measurement vectors (MMV) with different sensing matrices, where the signals of interest are assumed to be jointly sparse and each signal is sampled by its own sensing matrix in the presence of noise. Using mathematical tools, we develop upper and lower bounds of the failure probability of the support set reconstruction in terms of the sparsity, the ambient dimension, the minimum signal-to-noise ratio, the number of measurement vectors, and the number of measurements. These bounds can be used to provide guidelines for determining the system parameters for various compressed sensing applications with noisy MMV with different sensing matrices. Based on the bounds, we develop necessary and sufficient conditions for reliable support set reconstruction. We interpret these conditions to provide theoretical explanations regarding the benefits of taking more measurement vectors. We then compare our sufficient condition with the existing results for noisy MMV with the same sensing matrix. As a result, we show that noisy MMV with different sensing matrices may require fewer measurements for reliable support set reconstruction, under a sublinear sparsity regime in a low noise-level scenario.