稀疏恢复的可验证与可计算性能分析
Verifiable and computable performance analysis of sparsity recovery
AI总结：
本文针对任意感知矩阵定义了一族优化问题形式的优度度量，并设计了具有全局收敛保证的算法来计算这些度量，从而实现了稀疏恢复性能的可验证与可计算分析。
摘要：
在本文中，我们开发了稀疏恢复的可验证与可计算性能分析。我们针对任意感知矩阵定义了一族优度度量，将其表示为一组优化问题，并设计了具有理论全局收敛保证的算法来计算这些优度度量。所提出的算法求解一系列二阶锥规划或线性规划。作为副产品，我们实现了一个高效算法来验证无噪声情况下精确稀疏恢复的充分条件。我们根据这些优度度量推导了恢复误差的性能界。我们还分析证明，对于一大类随机感知矩阵，只要测量数量相对较大，所开发的优度度量是非退化的。数值实验表明，与基于受限等距性质的性能界相比，当信号的稀疏水平相对较低时，我们的误差界适用于更广泛的问题且更紧致。
英文摘要：
In this paper, we develop verifiable and computable performance analysis of sparsity recovery. We define a family of goodness measures for arbitrary sensing matrices as a set of optimization problems, and design algorithms with a theoretical global convergence guarantee to compute these goodness measures. The proposed algorithms solve a series of second-order cone programs, or linear programs. As a by-product, we implement an efficient algorithm to verify a sufficient condition for exact sparsity recovery in the noise-free case. We derive performance bounds on the recovery errors in terms of these goodness measures. We also analytically demonstrate that the developed goodness measures are non-degenerate for a large class of random sensing matrices, as long as the number of measurements is relatively large. Numerical experiments show that, compared with the restricted isometry based performance bounds, our error bounds apply to a wider range of problems and are tighter, when the sparsity levels of the signals are relatively low.
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