Title: Learning Filter Functions in Regularisers by Minimising Quotients,
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.