Sparse solution of underdetermined linear equations via adaptively iterative thresholding - Zeng, Jinshan; Lin, Shaobo; Xu, Zongben (Signal processing, 2014)
Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is mainly due to their low computational complexities, especially for large scale applications. The aim of this paper is to provide guarantees on the global convergence of a wide class of iterative thresholding algorithms. Since the thresholds of the considered algorithms are set adaptively at each iteration, we call them adaptively iterative thresholding (AIT) algorithms. As the main result, we show that as long as A satisfies a certain coherence property, AIT algorithms can find the correct support set within finite iterations, and then converge to the original sparse solution exponentially fast once the correct support set has been identified. Meanwhile, we also demonstrate that AIT algorithms are robust to the algorithmic parameters. In addition, it should be pointed out that most of the existing iterative thresholding algorithms such as hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are included in the class of AIT algorithms studied in this paper. •We provide the convergence analysis of a class of adaptively iterative thresholding (AIT) algorithms for sparse solution of underdetermined linear equations y=Ax.•AIT algorithm converges to the unique sparsest solution with a linearly asymptotic convergence rate under the assumption that A satisfies a certain coherence property.•AIT algorithm finds the correct support set within finite iterations.•Most of the commonly used iterative thresholding algorithms are included in the class of AIT algorithms studied in this paper.