16,523 research outputs found
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
Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-point Attracting Projection
A recursive algorithm named Zero-point Attracting Projection (ZAP) is
proposed recently for sparse signal reconstruction. Compared with the reference
algorithms, ZAP demonstrates rather good performance in recovery precision and
robustness. However, any theoretical analysis about the mentioned algorithm,
even a proof on its convergence, is not available. In this work, a strict proof
on the convergence of ZAP is provided and the condition of convergence is put
forward. Based on the theoretical analysis, it is further proved that ZAP is
non-biased and can approach the sparse solution to any extent, with the proper
choice of step-size. Furthermore, the case of inaccurate measurements in noisy
scenario is also discussed. It is proved that disturbance power linearly
reduces the recovery precision, which is predictable but not preventable. The
reconstruction deviation of -compressible signal is also provided. Finally,
numerical simulations are performed to verify the theoretical analysis.Comment: 29 pages, 6 figure
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