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Learning Filter Functions in Regularisers by Minimising Quotients

  • Martin Benning
  • Guy Gilboa
  • Joana Sarah GrahEmail author
  • Carola-Bibiane Schönlieb
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

Learning approaches have recently become very popular in the field of inverse problems. A large variety of methods has been established in recent years, ranging from bi-level learning to high-dimensional machine learning techniques. Most learning approaches, however, only aim at fitting parametrised models to favourable training data whilst ignoring misfit training data completely. In this paper, we follow up on the idea of learning parametrised regularisation functions by quotient minimisation as established in [3]. We extend the model therein to include higher-dimensional filter functions to be learned and allow for fit- and misfit-training data consisting of multiple functions. We first present results resembling behaviour of well-established derivative-based sparse regularisers like total variation or higher-order total variation in one-dimension. Our second and main contribution is the introduction of novel families of non-derivative-based regularisers. This is accomplished by learning favourable scales and geometric properties while at the same time avoiding unfavourable ones.

Keywords

Regularisation learning Non-linear eigenproblem Sparse regularisation Generalised inverse power method 

Notes

Acknowledgements.

MB acknowledges support from the Leverhulme Trust early career fellowship “Learning from mistakes: a supervised feedback-loop for imaging applications” and the Newton Trust. GG acknowledges support by the Israel Science Foundation (grant 718/15). JSG acknowledges support by the NIHR Cambridge Biomedical Research Centre. CBS acknowledges support from Leverhulme Trust project Breaking the non-convexity barrier, EPSRC grant EP/M00483X/1, EPSRC centre EP/N014588/1, the Cantab Capital Institute for the Mathematics of Information, and from CHiPS (Horizon 2020 RISE project grant).

Data Statement. The corresponding MATLAB\(^{\textregistered }\) code is publicly available on Apollo - University of Cambridge Repository (https://doi.org/10.17863/CAM.8419).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Martin Benning
    • 1
  • Guy Gilboa
    • 2
  • Joana Sarah Grah
    • 1
    Email author
  • Carola-Bibiane Schönlieb
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Electrical Engineering DepartmentTechnion - Israel Institute of TechnologyHaifaIsrael

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