# Learning Filter Functions in Regularisers by Minimising Quotients

## TL;DR

This paper introduces a method for learning regularisation functions in inverse problems by quotient minimisation, extending previous models to higher dimensions and multiple data types, and proposing novel non-derivative regularisers.

## Contribution

It extends quotient minimisation to include higher-dimensional filter functions and multiple training data types, and introduces new non-derivative regularisers.

## Key findings

- Behaves like total variation in 1D.
- Learns scales and geometric properties.
- Proposes novel non-derivative regularisers.

## 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.