Precisely Verifying the Null Space Conditions in Compressed Sensing: A Sandwiching Algorithm
This work addresses a computational bottleneck in compressed sensing for researchers and practitioners, though it is incremental as it builds on known challenges.
The paper tackles the problem of efficiently verifying the null space condition in compressed sensing by proposing new algorithms to compute exact values of α_k, with empirical results showing performance improvements and lower complexity than existing methods.
In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an $(n-m) \times n$ ($m>0$) CS matrix $A$ and a positive $k$, we are interested in computing $\displaystyle α_k = \max_{\{z: Az=0,z\neq 0\}}\max_{\{K: |K|\leq k\}}$ ${\|z_K \|_{1}}{\|z\|_{1}}$, 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 $α_k < {1}{2}$. However, computing $α_k$ is known to be extremely challenging. In this paper, we first propose a series of new polynomial-time algorithms to compute upper bounds on $α_k$. Based on these new polynomial-time algorithms, we further design a new sandwiching algorithm, to compute the \emph{exact} $α_k$ 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 $α_k$, with much lower complexity than exhaustive search.