Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

TL;DR

This paper investigates stochastic inverse problems involving probability distributions, focusing on stability, regularization, and Wasserstein gradient flows, highlighting the importance of metric choice in optimization within probability spaces.

Contribution

It introduces a framework for stochastic inverse problems using measure transport theory and analyzes how metric selection affects stability and optimization.

Findings

Metric choice influences stability and optimizer properties.

Regularization and gradient flows are adapted to probability spaces.

Tools from measure transport theory are essential for analysis.

Abstract

Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric -- both in the design of the loss function and in the optimization process -- significantly impacts the stability and properties of the optimizer.