Hard thresholding pursuit algorithms: Number of iterations

• Published in: Applied and computational harmonic analysis Vol. 41; no. 2; pp. 412 - 435
• Main Authors: Bouchot, Jean-Luc, Foucart, Simon, Hitczenko, Pawel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2016
• Subjects: Compressive sensing; Hard thresholding; Iterative algorithms; Nonuniform sparse recovery; Random measurements; Uniform sparse recovery; Uniform sparse recovery; Compressive sensing; 65F10; Hard thresholding; Random measurements; Iterative algorithms; 65J20; 94A12; Nonuniform sparse recovery; 15A29
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2016.03.002

The Hard Thresholding Pursuit algorithm for sparse recovery is revisited using a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered from compressive linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a certain restricted isometry condition. The recovery is also robust to measurement error. The same conclusions are derived for a variation of Hard Thresholding Pursuit, called Graded Hard Thresholding Pursuit, which is a natural companion to Orthogonal Matching Pursuit and runs without a prior estimation of the sparsity level. In addition, for two extreme cases of the vector shape, it is shown that, with high probability on the draw of random measurements, a fixed sparse vector is robustly recovered in a number of iterations precisely equal to the sparsity level. These theoretical findings are experimentally validated, too.