在压缩感知中精确验证空域条件：一种 Sandwiching 算法
Precisely Verifying the Null Space Conditions in Compressed Sensing: A Sandwiching Algorithm
AI总结：
本文提出新算法验证压缩感知中的空域条件，通过高效计算α_k，改进了传统方法的复杂度和精度。
摘要：
本文提出新的高效算法来验证压缩感知（CS）中的空域条件。给定一个(n-m)×n（m>0）的CS矩阵A和正数k，我们感兴趣于计算α_k = max{z: Az=0, z≠0} max{K: |K|≤k} ||z_K||₁||z||₁，其中K代表{1,2,...,n}的子集，|K|是K的基数。特别地，我们关注找到使得α_k < 1/2的最大k。然而，计算α_k被认为极具挑战性。本文首先提出一系列新的多项式时间算法来计算α_k的上界。基于这些新算法，我们进一步设计了一种新的Sandwiching算法，以大大降低复杂度的方式计算精确的α_k。当需要时，这种新的Sandwiching算法还能在计算复杂度和结果精度之间实现平滑的权衡。实验证明了我们的算法在性能上的改进；并且我们的算法输出精确的α_k值，其复杂度远低于穷举搜索。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
