# Bregman Itoh--Abe methods for sparse optimisation

## TL;DR

This paper introduces Bregman Itoh--Abe methods, a class of discrete gradient algorithms for sparse regularisation in inverse problems, offering guaranteed convergence and improved efficiency by leveraging problem structure.

## Contribution

It develops Bregman discrete gradient methods for inverse problems, providing convergence guarantees and demonstrating their effectiveness in sparse optimisation.

## Key findings

- Methods achieve rapid convergence rates.
- Algorithms are unconditionally dissipative.
- Numerical results show effectiveness in convex and non-convex cases.

## Abstract

In this paper we propose optimisation methods for variational regularisation problems based on discretising the inverse scale space flow with discrete gradient methods. Inverse scale space flow generalises gradient flows by incorporating a generalised Bregman distance as the underlying metric. Its discrete-time counterparts, Bregman iterations and linearised Bregman iterations, are popular regularisation schemes for inverse problems that incorporate a priori information without loss of contrast. Discrete gradient methods are tools from geometric numerical integration for preserving energy dissipation of dissipative differential systems. The resultant Bregman discrete gradient methods are unconditionally dissipative, and achieve rapid convergence rates by exploiting structures of the problem such as sparsity. Building on previous work on discrete gradients for non-smooth, non-convex optimisation, we prove convergence guarantees for these methods in a Clarke subdifferential framework. Numerical results for convex and non-convex examples are presented.