On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint

TL;DR

This paper proves that inverse problems with truncated quadratic regularization are NP-hard, highlighting the computational difficulty of extending Mumford-Shah type regularization to general inverse problems.

Contribution

It establishes the NP-hardness of solving or approximating inverse problems with truncated quadratic regularization, contrasting with polynomial-time solutions for the Mumford-Shah functional.

Findings

Truncated quadratic regularization problems are NP-hard to solve or approximate.

Polynomial-time solutions exist for the Mumford-Shah functional with identity operator.

The paper discusses the link between truncated quadratic minimization and sparsity constraints.

Abstract

We show that inverse problems with a truncated quadratic regularization are NP-hard in general to solve, or even approximate up to an additive error. This stands in contrast to the case corresponding to a finite-dimensional approximation to the Mumford-Shah functional, where the operator involved is the identity and for which polynomial-time solutions are known. Consequently, we confirm the infeasibility of any natural extension of the Mumford-Shah functional to general inverse problems. A connection between truncated quadratic minimization and sparsity-constrained minimization is also discussed.