Thresholding gradient methods in Hilbert spaces: support identification and linear convergence
Abstract
We study theℓ1regularized least squares optimization problem in a separable Hilbert space. We show that the iterative soft-thresholding algorithm (ISTA) converges linearly, without making any assumption on the linear operator into play or on the problem. The result is obtained combining two key concepts: the notion ofextended support, a finite set containing the support, and the notion ofconditioning over finite-dimensional sets. We prove that ISTA identifies the solution extended support after a finite number of iterations, and we derive linear convergence from the conditioning property, which is always satisfied forℓ1regularized least squares problems. Our analysis extends to the entire class of thresholding gradient algorithms, for which we provide a conceptually new proof of strong convergence, as well as convergence rates.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Jan 01, 2020
- Source ID
- 10.1051/cocv/2019011
Entities
People
- Guillaume Garrigos
- Lorenzo Rosasco
- Silvia Villa
Organizations
- European Research Council
- Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni
- United States Air Force