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On sliced Cramér metrics

William Leeb

TL;DR

This work analyzes sliced $p$-Cramér metrics, establishing stability under push-forward deformations and extensions to tomographic projections. It derives general deformation-growth bounds expressed via the maximum displacement $\varepsilon_ ext{∞}( ext{Φ})$ and mean mixed norms $\|f olinebreak ext{ } oldsymbol{p,r} ight $, and sharpens these bounds in key cases such as rotations, translations, and dilations. The paper also studies how these metrics behave under convolution, provides rigorous Fourier-based discretization error bounds for 1D and 2D cases, and proves robustness to additive heteroscedastic sub-Gaussian noise with provable convergence and concentration results. Numerical experiments corroborate the theoretical bounds on deformations, projections, and noise, highlighting the practical stability and noise resilience of sliced Cramér metrics in imaging-like tasks such as tomography and cryo-EM. Overall, the results position sliced Cramér distances as stable, discretizable, and noise-robust tools for comparing high-dimensional functions and tomographic projections with potential impact in imaging and signal processing.

Abstract

This paper studies the family of sliced Cramér metrics, quantifying their stability under distortions of the input functions. Our results bound the growth of the sliced Cramér 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 Cramér distances between tomographic projections. In addition, we remark on the effect of convolution on the sliced Cramér 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.

On sliced Cramér metrics

TL;DR

This work analyzes sliced -Cramér metrics, establishing stability under push-forward deformations and extensions to tomographic projections. It derives general deformation-growth bounds expressed via the maximum displacement and mean mixed norms , and sharpens these bounds in key cases such as rotations, translations, and dilations. The paper also studies how these metrics behave under convolution, provides rigorous Fourier-based discretization error bounds for 1D and 2D cases, and proves robustness to additive heteroscedastic sub-Gaussian noise with provable convergence and concentration results. Numerical experiments corroborate the theoretical bounds on deformations, projections, and noise, highlighting the practical stability and noise resilience of sliced Cramér metrics in imaging-like tasks such as tomography and cryo-EM. Overall, the results position sliced Cramér distances as stable, discretizable, and noise-robust tools for comparing high-dimensional functions and tomographic projections with potential impact in imaging and signal processing.

Abstract

This paper studies the family of sliced Cramér metrics, quantifying their stability under distortions of the input functions. Our results bound the growth of the sliced Cramér 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 Cramér distances between tomographic projections. In addition, we remark on the effect of convolution on the sliced Cramér 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.

Paper Structure

This paper contains 59 sections, 46 theorems, 281 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Geometric deformations
  3. Convolutions
  4. Discretizations and additive noise
  5. Notation and conventions
  6. Lebesgue norms
  7. Absolute continuity
  8. The Fourier transform
  9. $C^r$ functions
  10. Tomographic projections and the Radon transform
  11. Push-forwards
  12. Sub-Gaussian random variables
  13. Outline of the remainder of the paper
  14. Background and preliminary results
  15. Deformations and displacement
  16. ...and 44 more sections

Key Result

Proposition 2.1

Let $x^* = \operatorname{arg\,max}_x |x - \Psi(x)|$, and let $u^* = (x^* - \Psi(x^*)) / |x^* - \Psi(x^*)|$. Then

Figures (8)

  • Figure 1: The function $f$ described in Section \ref{['sec:numerical_deformations']}, and examples of the deformations applied to $f$. From left to right: the original function; a translation; a rotation; a dilation.
  • Figure 2: Results of the experiment described in Section \ref{['sec:numerical_deformations']}, showing the distances from $f$ to its translations as a function of the translation size.
  • Figure 3: Results of the experiment described in Section \ref{['sec:numerical_deformations']}, showing the distances from $f$ to its rotations as a function of the rotation angle.
  • Figure 4: Results of the experiment described in Section \ref{['sec:numerical_deformations']}, showing the distances from $f$ to its dilations as a function of the dilation size.
  • Figure 5: Projections used in the experiment described in Section \ref{['sec:numerical_rotations3d']}. Left: the projection onto the $xy$-plane of the original spiral function of $f$. Right: the projection onto the $xy$-plane of a rotation of $f$ within the $yz$-plane.
  • ...and 3 more figures

Theorems & Definitions (94)

  • Remark 1
  • Proposition 2.1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 84 more