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.

Let X be a smooth, projective, geometrically connected curve over a finite field Fq, and let G be a split semisimple algebraic group over Fq. Its dual group Gˆ is a split reductive group over Z. Conjecturally, any l-adic Gˆ-local system on X (equivalently, any conjugacy class of continuous homomorphisms π1(X)→Gˆ(Q¯¯¯¯l)) should be associated with an everywhere unramified automorphic representation of the group G. We show that for any homomorphism π1(X)→Gˆ(Q¯¯¯¯l) of Zariski dense image, there exists a finite Galois cover Y→X over which the associated local system becomes automorphic.

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