FAASTA: A fast solver for total-variation regularization of ill-conditioned problems with application to brain imaging

TL;DR

FASTA is a fast algorithm for solving total variation regularization problems, especially in ill-conditioned brain imaging applications, by balancing gradient and proximal step costs to improve convergence speed.

Contribution

It introduces fAASTA, a variant of FISTA that adaptively refines the TV proximal operator's tolerance to enhance efficiency in challenging inverse problems.

Findings

fAASTA reduces computation time in brain imaging tasks.

The method improves convergence speed for non-exact proximal operators.

Empirical benchmarks demonstrate efficiency gains over existing methods.

Abstract

The total variation (TV) penalty, as many other analysis-sparsity problems, does not lead to separable factors or a proximal operatorwith a closed-form expression, such as soft thresholding for the $\ell\_1$ penalty. As a result, in a variational formulation of an inverse problem or statisticallearning estimation, it leads to challenging non-smooth optimization problemsthat are often solved with elaborate single-step first-order methods. When thedata-fit term arises from empirical measurements, as in brain imaging, it isoften very ill-conditioned and without simple structure. In this situation, in proximal splitting methods, the computation cost of thegradient step can easily dominate each iteration. Thus it is beneficialto minimize the number of gradient steps.We present fAASTA, a variant of FISTA, that relies on an internal solver forthe TV proximal operator, and refines its tolerance to balance computationalcost of the gradient and the proximal steps. We give benchmarks andillustrations on "brain decoding": recovering brain maps from noisymeasurements to predict observed behavior. The algorithm as well as theempirical study of convergence speed are valuable for any non-exact proximaloperator, in particular analysis-sparsity problems.