Precisely Verifying the Null Space Conditions in Compressed Sensing: A Sandwiching Algorithm

The null space condition of sensing matrices plays an important role in guaranteeing the success of compressed sensing. In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an (n − m) × n (m > 0) CS matrix A and a positive k, we are interested in computing αk = max {z:Az=0,z ̸=0} max {K:|K|≤k} ∥zK∥1 ∥z∥1 , where K represents subsets of {1, 2, ..., n}, and |K| is the cardinality of K. In particular, we are interested in finding the maximum k such that αk < 12 . However, computing αk is known to be extremely challenging. In this paper, we first propose a series of new polynomial-time algorithms to compute upper bounds on αk . Based on these new polynomial-time algorithms, we further design a new sandwiching algorithm, to compute the exact αk with greatly reduced complexity. When needed, this new sandwiching algorithm also achieves a smooth tradeoff between computational complexity and result accuracy. Empirical results show the performance improvements of our algorithm over existing known methods; and our algorithm outputs precise values of αk , with much lower complexity than exhaustive search.