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

TL;DR

This paper introduces efficient algorithms to verify null space conditions in compressed sensing, enabling precise computation of key parameters with reduced complexity compared to exhaustive methods.

Contribution

The paper presents new polynomial-time algorithms for upper bounding and an innovative sandwiching algorithm for exact null space condition verification in compressed sensing.

Findings

Algorithms outperform existing methods in empirical tests.

The sandwiching algorithm achieves accurate results with lower complexity.

Exact null space condition values are computed efficiently.

Abstract

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 $\displaystyle \alpha_k = \max_{\{z: Az=0,z\neq 0\}}\max_{\{K: |K|\leq k\}}$ ${\|z_K \|_{1}}{\|z\|_{1}}$, 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 $\alpha_k < {1}{2}$. However, computing $\alpha_k$ is known to be extremely challenging. In this paper, we first propose a series of new polynomial-time algorithms to compute upper bounds on $\alpha_k$. Based on these new polynomial-time algorithms, we further design a new sandwiching algorithm, to compute the \emph{exact} $\alpha_k$ 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 $\alpha_k$, with much lower complexity than exhaustive search.