A sparse optimization approach to infinite infimal convolution regularization

Kristian Bredies, Marcello Carioni*, Martin Holler, Yury Korolev, Carola Bibiane Schönlieb

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In this paper we introduce the class of infinite infimal convolution functionals and apply these functionals to the regularization of ill-posed inverse problems. The proposed regularization involves an infimal convolution of a continuously parametrized family of convex, positively one-homogeneous functionals defined on a common Banach space X. We show that, under mild assumptions, this functional admits an equivalent convex lifting in the space of measures with values in X. This reformulation allows us to prove well-posedness of a Tikhonov regularized inverse problem and opens the door to a sparse analysis of the solutions. In the case of finite-dimensional measurements we prove a representer theorem, showing that there exists a solution of the inverse problem that is sparse, in the sense that it can be represented as a linear combination of the extremal points of the ball of the lifted infinite infimal convolution functional. Then, we design a generalized conditional gradient method for computing solutions of the inverse problem without relying on an a priori discretization of the parameter space and of the Banach space X. The iterates are constructed as linear combinations of the extremal points of the lifted infinite infimal convolution functional. We prove a sublinear rate of convergence for our algorithm and apply it to denoising of signals and images using, as regularizer, infinite infimal convolutions of fractional-Laplacian-type operators with adaptive orders of smoothness and anisotropies.

Original languageEnglish
JournalNumerische Mathematik
DOIs
Publication statusE-pub ahead of print/First online - 21 Nov 2024

Keywords

  • UT-Hybrid-D
  • 2024 OA procedure
  • 65J20
  • 65K10
  • 35R11
  • 49J45

Fingerprint

Dive into the research topics of 'A sparse optimization approach to infinite infimal convolution regularization'. Together they form a unique fingerprint.

Cite this