Jump-sparse and sparse recovery using Potts functionals

TL;DR

This paper introduces a novel approach using inverse Potts functionals for recovering jump-sparse and sparse signals from incomplete, noisy, and blurred data, with analytical insights and an efficient optimization algorithm.

Contribution

It provides the first analytical results on inverse Potts functionals, develops a new optimization method combining dynamic programming and ADMM, and demonstrates superior performance in signal reconstruction.

Findings

The method effectively reconstructs jump-sparse signals from noisy, incomplete data.

It outperforms classical approaches like TV minimization and recent sparse recovery algorithms.

Experiments confirm the method's robustness and accuracy in various scenarios.

Abstract

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $\ell^1$ minimization (sparse signals).