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.
Highlights
This article is dedicated to Mila Nikolova whose keen interest and inspiring discussions have encouraged our research on geometric integration for optimisation
We study the Itoh–Abe discrete gradient method applied to the inverse scale space (ISS) flow
We review papers on discrete gradient methods for optimisation based on gradient flows
Summary
This article is dedicated to Mila Nikolova whose keen interest and inspiring discussions have encouraged our research on geometric integration for optimisation. We propose and study optimisation schemes by using tools from geometric numerical integration to solve the inverse scale space (ISS) flow. The ISS flow is a dissipative system, and its dissipative structure is determined by the function J This allows one to solve (1.1) while incorporating a priori information into the optimisation scheme, with the benefits of converging to superior solutions, and doing so faster. We propose to discretise the inverse scale space flow with discrete gradient methods These are methods from geometric numerical integration that preserve the aforementioned geometric structures in a general setting. The theoretical convergence rates of the discrete gradient methods match those of explicit gradient descent and coordinate descent [21] The drawback of these methods is that the updates are in general implicit.
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