The quadratic Wasserstein metric for inverse data matching

TL;DR

This paper explores the quadratic Wasserstein ($W_2$) distance's dual effects in inverse problems, showing its noise-robust smoothing in infinite dimensions and improved convexity in finite-dimensional optimization, enhancing data matching techniques.

Contribution

It analytically and numerically characterizes how the $W_2$ distance improves robustness and convexity in inverse data matching problems, offering insights into its advantages over traditional metrics.

Findings

$W_2$ distance smooths inverse solutions, reducing sensitivity to high-frequency noise.

In finite dimensions, $W_2$ leads to more convex optimization problems.

$W_2$ enhances robustness and optimization stability in inverse data matching.

Abstract

This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that for some finite-dimensional problems, the $W_2$ distance leads to optimization problems that have better convexity than the classical $L^2$ and $H^{-1}$ distances, making it a more preferred distance to use when solving such inverse matching problems.