Linear Convergence of Adaptively Iterative Thresholding Algorithms for Compressed Sensing

This paper studies the convergence of the adaptively iterative thresholding (AIT) algorithm for compressed sensing. We first introduce a generalized restricted isometry property (gRIP). Then we prove that the AIT algorithm converges to the original sparse solution at a linear rate under a certain gRIP condition in the noise free case. While in the noisy case, its convergence rate is also linear until attaining a certain error bound. Moreover, as by-products, we also provide some sufficient conditions for the convergence of the AIT algorithm based on the two well-known properties, i.e., the coherence property and the restricted isometry property (RIP), respectively. It should be pointed out that such two properties are special cases of gRIP. The solid improvements on the theoretical results are demonstrated and compared with the known results. Finally, we provide a series of simulations to verify the correctness of the theoretical assertions as well as the effectiveness of the AIT algorithm.Comment: 15 pages, 5 figure

New Bounds for Restricted Isometry Constants

In this paper we show that if the restricted isometry constant $\delta_k$ of the compressed sensing matrix satisfies $$\delta_k < 0.307,$$ then $k$-sparse signals are guaranteed to be recovered exactly via $\ell_1$ minimization when no noise is present and $k$-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantively improved. An explicitly example is constructed in which $\delta_{k}=\frac{k-1}{2k-1} < 0.5$, but it is impossible to recover certain $k$-sparse signals