结合几何与组合：稀疏信号恢复的统一方法
Combining geometry and combinatorics: A unified approach to sparse signal recovery
AI总结：
本文提出一种统一方法，通过高质量不平衡扩展图的邻接矩阵，将几何与组合两类稀疏信号恢复算法联系起来，并推广了受限等距性质到l_p范数，从而得到新的确定性测量矩阵构造和信号恢复算法。
摘要：
稀疏信号恢复主要有两种算法方法：几何方法和组合方法。几何方法从测量矩阵的几何约束出发，然后使用线性规划从其测量中解码关于信号的信息。组合方法构造测量矩阵和匹配的组合解码算法。我们提出了这两种稀疏信号恢复算法的统一方法。统一元素是高质量不平衡扩展图的邻接矩阵。我们将对信号恢复至关重要的压缩感知结果中的受限等距性质（RIP）从欧几里得范数推广到p约等于1的l_p范数，然后证明不平衡扩展图本质上等价于RIP-p矩阵。从这些矩阵的已知确定性构造中，我们获得了新的确定性测量矩阵构造和信号恢复算法，与先前的确定性算法相比，在测量数量或噪声容忍度方面更优。
英文摘要：
There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix and then uses linear programming to decode information about the signal from its measurements. The combinatorial approach constructs the measurement matrix and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of high-quality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the l_p norm for p about 1, and then show that unbalanced expanders are essentially equivalent to RIP-p matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance.
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