Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On metrics robust to noise and deformations

William Leeb

TL;DR

This work introduces Volterra $p$-norms, $\|f\|_{V^p}$, and the induced Volterra $p$-distances as a robust class of integral probability metrics for univariate functions and their tomographic projections. Through a variational characterization and a suite of robustness results, the authors show that these distances are resistant to a broad class of deformations and additive Gaussian noise, with stronger bounds in certain monotone or rotated-projection settings. The paper extends these ideas to sliced Volterra distances for multivariate functions, provides a trapezoidal-rule-based discrete framework with provable convergence rates ($O(1/n)$ for Lipschitz and $O(1/n^2)$ for smooth) and noise robustness, and demonstrates numerical behavior that parallels, and in some regimes outperforms, Wasserstein-based metrics. The results offer practical tools for robust comparison of signals and projections in applications such as tomography and cryo-EM, where data are often noisy and deformations are common.

Abstract

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

On metrics robust to noise and deformations

TL;DR

This work introduces Volterra -norms, , and the induced Volterra -distances as a robust class of integral probability metrics for univariate functions and their tomographic projections. Through a variational characterization and a suite of robustness results, the authors show that these distances are resistant to a broad class of deformations and additive Gaussian noise, with stronger bounds in certain monotone or rotated-projection settings. The paper extends these ideas to sliced Volterra distances for multivariate functions, provides a trapezoidal-rule-based discrete framework with provable convergence rates ( for Lipschitz and for smooth) and noise robustness, and demonstrates numerical behavior that parallels, and in some regimes outperforms, Wasserstein-based metrics. The results offer practical tools for robust comparison of signals and projections in applications such as tomography and cryo-EM, where data are often noisy and deformations are common.

Abstract

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

Paper Structure

This paper contains 42 sections, 22 theorems, 230 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Lebesgue $p$-norms
  4. Trapezoidal rule approximation
  5. Absolute continuity
  6. The Volterra operator
  7. Push-forwards and $\epsilon$-deformations
  8. Tomographic projections and the Radon transform
  9. Wasserstein distances
  10. Sliced Wasserstein distance
  11. Volterra distances
  12. The Volterra norms and distances
  13. Variational formulation of $\|f\|_{V^p}$
  14. Sliced Volterra distances
  15. Trapezoidal rule approximation to $\|f\|_{V^p}$
  16. ...and 27 more sections

Key Result

Theorem 2.1

If $F$ and $G$ are absolutely continuous functions on $[a,b]$, then

Figures (18)

  • Figure 1: The function \ref{['eq:transl_bump']} (far left, in blue) and its translations, used in the experiment from Section \ref{['sec:transl']}.
  • Figure 2: The first row shows the approximated Volterra distances (based on $n=500$ subintervals) between the function \ref{['eq:transl_bump']} and its shifts, as a function of the shift size. The second row shows the approximated Wasserstein distances, and the third row shows the approximated Lebesgue distances. The values of $p$ (from left to right) are $p=1,2, \infty$. See Section \ref{['sec:transl']} for details.
  • Figure 3: The function \ref{['eq:dilation_func']} (in blue) and its dilations, used in the experiment from Section \ref{['sec:dilations']}.
  • Figure 4: The first row shows the approximated Volterra distances (based on $n=500$ subintervals) between the function \ref{['eq:dilation_func']} and its dilates, as a function of the deformation size. The second row shows the approximated Wasserstein distances, and the third row shows the approximated Lebesgue distances. The values of $p$ (from left to right) are $p=1,2, \infty$. See Section \ref{['sec:dilations']} for details.
  • Figure 5: The function \ref{['eq:powers_func']} (in blue) and its deformations, used in the experiment from Section \ref{['sec:powers']}.
  • ...and 13 more figures

Theorems & Definitions (38)

  • Theorem 2.1: Integration by parts
  • Theorem 2.2
  • proof
  • Remark 1
  • Remark 2
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:variational']}
  • Theorem 4.1
  • Remark 3
  • Theorem 4.2
  • ...and 28 more