Decoupled Solution for Composite Sparse-plus-Smooth Inverse Problems

TL;DR

This paper introduces a decoupled optimization framework for inverse problems involving a sum of sparse and smooth components, enabling efficient reconstruction with improved accuracy.

Contribution

It develops a novel decoupling approach and a composite representer theorem for inverse problems with sparse and smooth parts, facilitating separate regularization and solution.

Findings

Decoupled algorithm improves signal reconstruction accuracy.

The composite model effectively separates sparse and smooth components.

Significant temporal gain demonstrated in Dirac recovery experiments.

Abstract

We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of our work is to study the case where one component is regularized with an atomic norm over a Banach space, which is known to promote sparse reconstruction, while the other is regularized with a quadratic norm over a Hilbert space, which promotes smooth solution. We show how this composite optimization problem can be reduced to an optimization problem over the Banach space component only up to a linear problem. This reveals a decoupling between the two components, allowing for a new composite representer theorem. It naturally induces a decoupled numerical procedure to solve the composite optimization problem. We exemplify our main result with a composite deconvolution problem of Dirac recovery over a smooth background. In this setting, we illustrate the relevance of a composite model and show a significant temporal gain on signal reconstruction, which results from our decoupled algorithmic approach.