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Bregman Itoh–Abe Methods for Sparse Optimisation

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Benning, Martin 
Riis, Erlend Skaldehaug  ORCID logo
Schönlieb, Carola-Bibiane 


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.



Article, Non-convex optimisation, Non-smooth optimisation, Bregman iteration, Inverse scale space, Geometric numerical integration, Discrete gradient methods, 49M37, 49Q15, 65K10, 90C26

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Journal of Mathematical Imaging and Vision

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Springer US
Leverhulme Trust (2016-611)
Horizon 2020 (RISE)
Horizon 2020 (RISE)
Engineering and Physical Sciences Research Council (GB) (EP/M00483X/1, EP/N014588/1)