On sliced Cram\'er metrics

TL;DR

This paper explores the stability and robustness of sliced Cramér metrics under various distortions, deformations, and noise, providing bounds, extensions, and efficient discretizations with numerical validation.

Contribution

It introduces bounds on sliced Cramér distances under geometric deformations, extends results to tomographic projections, and analyzes Fourier-based discretizations for robustness.

Findings

Bounds the growth of sliced Cramér distance under deformations

Extends analysis to tomographic projections

Demonstrates robustness of Fourier discretizations to noise

Abstract

This paper studies the family of sliced Cram\'er metrics, quantifying their stability under distortions of the input functions. Our results bound the growth of the sliced Cram\'er distance between a function and its geometric deformation by the product of the deformation's displacement size and the function's mean mixed norm. These results extend to sliced Cram\'er distances between tomographic projections. In addition, we remark on the effect of convolution on the sliced Cram\'er metrics. We also analyze efficient Fourier-based discretizations in 1D and 2D, and prove that they are robust to heteroscedastic noise. The results are illustrated by numerical experiments.