New algorithms for verifying the null space conditions in compressed sensing

The null space condition is a condition under which k-sparse signal can be recovered uniquely in compressed sensing (CS) problems. In this paper, new efficient algorithms are introduced to verify the null space condition for l
₁
minimization in compressed sensing. Suppose A is an (n - m) × n (m > 0) sensing matrix, we can verify whether the sensing matrix A satisfies the null space condition or not for k-sparse signals by computing α
_k
= max
_{{z: Az=0, z≠0}}
max
_{K:|K|≤k}
||z
_K
||1/||z||1, where K represents subsets of {1, 2,..., n}, and |K| is the cardinality of K. However, computing α
_k
is known to be extremely challenging because of high computational complexity. In this paper, a series of new polynomial-time algorithms are proposed to compute upper bounds on α
_k
. Based on these new polynomial-time algorithms, we further design new algorithm, which is called the sandwiching algorithm, to compute the exact α
_k
with much lower complexity than exhaustive search.