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Fisher-Rao distances between finite-energy signals in Gaussian noise

Franck Florin

TL;DR

This paper addresses the problem of quantifying dissimilarity between finite-energy signals observed within a bandwidth by mapping signals to distributions on a statistical manifold and using the Fisher-Rao distance under a complex Gaussian observation model $p(\boldsymbol{x}|\boldsymbol{\xi})$. It develops two parametric signal models, the full $L^2(\mathcal{B})$ manifold and the submanifold $L^2(\mathcal{B},\alpha)$ with known magnitude spectrum, and derives closed-form Fisher-Rao distances for both cases by solving the corresponding geodesic equations. A key theoretical contribution is the LDG theorem, which shows a linear dependence of magnitude and phase gradient dynamics along geodesics, and a practical finding that the submanifold is not fully geodesic, with distances on the submanifold exceeding those on the full manifold under most conditions. The results illuminate how magnitude and phase spectra, as well as SNR and bandwidth, control the geometric shaping of signal-space distances, with potential applications to signal databases, clustering, and physics-informed estimation or neural networks.

Abstract

This paper proposes representing finite-energy signals observed within a given bandwidth as parameters of a probability distribution and employing the information-geometric framework to compute the Fisher-Rao distance between these signals, considered as distributions.

Fisher-Rao distances between finite-energy signals in Gaussian noise

TL;DR

This paper addresses the problem of quantifying dissimilarity between finite-energy signals observed within a bandwidth by mapping signals to distributions on a statistical manifold and using the Fisher-Rao distance under a complex Gaussian observation model $p(\boldsymbol{x}|\boldsymbol{\xi})$. It develops two parametric signal models, the full $L^2(\mathcal{B})$ manifold and the submanifold $L^2(\mathcal{B},\alpha)$ with known magnitude spectrum, and derives closed-form Fisher-Rao distances for both cases by solving the corresponding geodesic equations. A key theoretical contribution is the LDG theorem, which shows a linear dependence of magnitude and phase gradient dynamics along geodesics, and a practical finding that the submanifold is not fully geodesic, with distances on the submanifold exceeding those on the full manifold under most conditions. The results illuminate how magnitude and phase spectra, as well as SNR and bandwidth, control the geometric shaping of signal-space distances, with potential applications to signal databases, clustering, and physics-informed estimation or neural networks.

Abstract

This paper proposes representing finite-energy signals observed within a given bandwidth as parameters of a probability distribution and employing the information-geometric framework to compute the Fisher-Rao distance between these signals, considered as distributions.

Paper Structure

This paper contains 28 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. How to compute the Fisher-Rao distance
  3. Notational conventions
  4. Methodology to compute the Fisher-Rao distance
  5. The case of the multivariate normal distributions
  6. Fisher-Rao distance in a submanifold
  7. Signal parametric modelling
  8. Observation of a finite-energy signal
  9. Signal dependence on parameters and manifold definitions
  10. The finite-energy signals and the manifold $L2(\mathcal{B})$
  11. The finite-energy signals with a known magnitude spectrum and the manifold $L2(\mathcal{B},\alpha)$
  12. General expressions of the Fisher-Rao distances
  13. Geodesic equations on the manifold $L2(\mathcal{B})$
  14. Fisher-Rao distance on the manifold $L2(\mathcal{B})$
  15. Geodesic equations on the submanifold $L2(\mathcal{B},\xi)$
  16. ...and 13 more sections

Key Result

Proposition 1

The model of finite-energy signals with a known magnitude spectrum verifies the regularity conditions, proving that $L2(\mathcal{B},\alpha)$ is a manifold.

Theorems & Definitions (29)

  • Remark 1
  • Proposition 1
  • proof
  • proof : Proof
  • proof
  • Theorem 2: Linearly Dependant Gradients (LDG) theorem
  • proof
  • Remark 2
  • Remark 3: LDG theorem significance
  • Proposition 3
  • ...and 19 more