Sparsity for dynamic inverse problems on Wasserstein curves with bounded variation

TL;DR

This paper extends static regularization techniques like total variation to dynamic inverse problems using Wasserstein-1 distance, allowing for discontinuous BV curves and demonstrating numerical implementability.

Contribution

It introduces a novel dynamic inverse problem framework with Wasserstein-1 regularization and proves existence, characterization, and numerical methods for sparse BV solutions.

Findings

Existence and characterization of sparse solutions in the dynamic setting.

Numerical implementation of BV curves in Wasserstein-1 space.

Experimental validation of the proposed method.

Abstract

We investigate a dynamic inverse problem using a regularization which implements the so-called Wasserstein-$1$ distance. It naturally extends well-known static problems such as lasso or total variation regularized problems to a (temporally) dynamic setting. Further, the decision variables, realized as BV curves, are allowed to exhibit discontinuities, in contrast to the design variables in classical optimal transport based regularization techniques. We prove the existence and a characterization of a sparse solution. Further, we use an adaption of the fully-corrective generalized conditional gradient method to experimentally justify that the determination of BV curves in the Wasserstein-$1$ space is numerically implementable.