On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint
Establishes computational hardness for a class of regularization problems, impacting researchers in inverse problems and optimization.
The paper proves that inverse problems with truncated quadratic regularization are NP-hard to solve or approximate, unlike the identity-operator case. This shows that extending Mumford-Shah functionals to general inverse problems is infeasible.
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.