具有限制isometry性质的确定性矩阵之路
The road to deterministic matrices with the restricted isometry property
摘要：
限制等距属性（RIP）是一个众所周知的矩阵条件，它为压缩感知提供了最先进的重建保证。虽然已知随机矩阵以高概率满足此性质，但确定性构造的成功率较低。在本文中，我们考虑各种技术证明RIP确定性，一些流行的和一些新颖的，我们评估其性能。在评估某些技术时，我们应用随机矩阵理论，无意中发现了一个简单的替代证明，即某些随机矩阵是RIP。后来，我们提出了一类特殊的矩阵作为RIP的候选者，即等角紧帧（ETF）。利用已知的真实ETF和强正则图之间的对应关系，我们研究了真实ETF是RIP的某些组合含义。具体来说，我们给出了概率直觉的一个新的界限上的团数的Paley图的素数阶，我们猜想，相应的ETF是RIP在一种类似于随机矩阵的方式。
英文摘要：
The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.
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。
