Thresholding gradient methods in Hilbert spaces: support identification and linear convergence

TL;DR

This paper proves linear convergence of the iterative soft-thresholding algorithm (ISTA) for -regularized least squares in Hilbert spaces, using support identification and conditioning concepts, applicable to a broad class of thresholding gradient methods.

Contribution

It establishes linear convergence of ISTA without assumptions on the operator, introduces the extended support concept, and extends results to all thresholding gradient algorithms.

Findings

ISTA identifies the extended support after finite iterations.

ISTA converges linearly without assumptions on the linear operator.

The analysis applies to all thresholding gradient algorithms, providing new convergence proofs.

Abstract

We study $\ell^1$ regularized 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 of extended support, a finite set containing the support, and the notion of conditioning 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 for $\ell^1$ regularized least squares problems. Our analysis extends to the the entire class of thresholding gradient algorithms, for which we provide a conceptually new proof of strong convergence, as well as convergence rates.