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Signals as submanifolds, and configurations of points

Tatyana Barron, Spencer Kelly, Colin Poulton

TL;DR

The paper develops a coordinate-free framework in which signals are modeled as submanifolds $M$ of a Riemannian ambient, with energies $E_1$ and $E_2$ capturing their capacity, and investigates energy bounds across concrete geometries. It derives upper and lower energy bounds for 1D signals in the Gaussian manifold $\mathcal{I}=\Omega\times P_n$ and for signals in configuration spaces $C_n(M)$, including explicit inequalities such as $E_1\le \rho(A,B)^2$, $E_2\le \rho(A,B)^3$, and $E_2\ge \tfrac{1}{3}\|B-A\|_3^3$ under suitable conditions. A key contribution is the relative ratio variance function, a quantitative measure of how a fixed graph embedding distorts the intrinsic metric when realized as a configuration in a manifold, and its invariance properties under uniform metric scaling. By treating time as potentially multi-dimensional and by linking energy, noise, and Fourier-like concepts within a geometric setting, the work aims to unify signal-processing intuition with the geometry of ambient spaces, enabling new analyses for systems of many agents or configurations in physical and social contexts.

Abstract

For the purposes of abstract theory of signal propagation, a signal is a submanifold of a Riemannian manifold. We obtain energy inequalities, or upper bounds, lower bounds on energy in a number of specific cases, including parameter spaces of Gaussians and spaces of configurations of points. We discuss the role of time as well as graph embeddings.

Signals as submanifolds, and configurations of points

TL;DR

The paper develops a coordinate-free framework in which signals are modeled as submanifolds of a Riemannian ambient, with energies and capturing their capacity, and investigates energy bounds across concrete geometries. It derives upper and lower energy bounds for 1D signals in the Gaussian manifold and for signals in configuration spaces , including explicit inequalities such as , , and under suitable conditions. A key contribution is the relative ratio variance function, a quantitative measure of how a fixed graph embedding distorts the intrinsic metric when realized as a configuration in a manifold, and its invariance properties under uniform metric scaling. By treating time as potentially multi-dimensional and by linking energy, noise, and Fourier-like concepts within a geometric setting, the work aims to unify signal-processing intuition with the geometry of ambient spaces, enabling new analyses for systems of many agents or configurations in physical and social contexts.

Abstract

For the purposes of abstract theory of signal propagation, a signal is a submanifold of a Riemannian manifold. We obtain energy inequalities, or upper bounds, lower bounds on energy in a number of specific cases, including parameter spaces of Gaussians and spaces of configurations of points. We discuss the role of time as well as graph embeddings.
Paper Structure (11 sections, 8 theorems, 95 equations, 3 figures)

This paper contains 11 sections, 8 theorems, 95 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Signals
  3. Energy
  4. More about energy
  5. Noise, Fourier transform
  6. Relativity considerations and time
  7. Parameter spaces of Gaussians
  8. Configurations of points
  9. Riemannian metric
  10. Energy
  11. Graph embedddings and relative ratio variance function

Key Result

Theorem 2.13

Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold. $(i)$ Let $\gamma:[0,1]\to \tilde{M}$ be a one dimensional path from $p=\gamma(0)$ to $q=\gamma(1)$. For the signal $(M=\gamma([0,1]),g,p,q)$ $(ii)$ Let $k\ge 2$. Let $k,a,b,\Lambda,\gamma,p,q$ define a $k$-dimensional path from $p$ to $q$ in $\tilde{M}$, as in Definition def:kpath. For the signal $(M=\gamma(\Lambda),g,p,q)$

Figures (3)

  • Figure 1: $(M,g,A,B)$ in Example \ref{['exm:lineseg']}.
  • Figure 2: $y=f(x)$, $0\le x\le 3$
  • Figure 3: A $2$-dimensional path from $p$ to $q$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 16 more