A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems

TLDR

Iterative shrinkage‑thresholding algorithms (ISTA) are simple, scalable methods for linear inverse problems in signal and image processing, yet they are known to converge slowly. This work introduces FISTA, a fast iterative shrinkage‑thresholding algorithm that retains ISTA’s simplicity while achieving a significantly faster global convergence rate. FISTA achieves this by modifying ISTA’s update rule to accelerate convergence while keeping per‑iteration complexity unchanged. Numerical experiments on wavelet‑based image deblurring show that FISTA is several orders of magnitude faster than ISTA.

Abstract

We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude.