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Geometric signals

Tatyana Barron

TL;DR

The paper addresses modeling information transmission with a geometric paradigm: signals are defined as relative cobordisms between boundary manifolds within a Riemannian setting, rather than as time-based functions. It introduces a Fourier-transform-like self-map on cobordisms and defines energy via $E(M_g)=\int_M f_A(x)dV_g(x)$, with the transformed energy $E(F(M_g))=\int_M f_X(x)dV_g(x)$, enabling a quantitative comparison of signal and transformed-signal strength. The work also develops operations such as noise (local metric deformations), filtering, and composition of signals, and proves energy-inequality theorems (Theorems thmain1 and thmain2) that describe how energy behaves under transformation and combination. By connecting geometric data analysis concepts to information transmission, the paper offers a foundational step toward a geometry-based information framework with potential relevance to neural networks and EM data analysis.

Abstract

In signal processing, a signal is a function. Conceptually, replacing a function by its graph, and extending this approach to a more abstract setting, we define a signal as a submanifold M of a Riemannian manifold (with corners) that satisfies additional conditions. In particular, it is a relative cobordism between two manifolds with boundaries. We define energy as the integral of the distance function to the first of these boundary manifolds. Composition of signals is composition of cobordisms. A "time variable" can appear explicitly if it is explictly given (for example, if the manifold is of the form $Σ\times [0,1]$). Otherwise, there is no designated "time dimension", although the cobordism may implicitly indicate the presence of dynamics. We interpret a local deformation of the metric as noise. The assumptions on M allow to define a map $M\to M$ that we call a Fourier transform. We prove inequalities that illustrate the properties of energy of signals in this setting.

Geometric signals

TL;DR

The paper addresses modeling information transmission with a geometric paradigm: signals are defined as relative cobordisms between boundary manifolds within a Riemannian setting, rather than as time-based functions. It introduces a Fourier-transform-like self-map on cobordisms and defines energy via , with the transformed energy , enabling a quantitative comparison of signal and transformed-signal strength. The work also develops operations such as noise (local metric deformations), filtering, and composition of signals, and proves energy-inequality theorems (Theorems thmain1 and thmain2) that describe how energy behaves under transformation and combination. By connecting geometric data analysis concepts to information transmission, the paper offers a foundational step toward a geometry-based information framework with potential relevance to neural networks and EM data analysis.

Abstract

In signal processing, a signal is a function. Conceptually, replacing a function by its graph, and extending this approach to a more abstract setting, we define a signal as a submanifold M of a Riemannian manifold (with corners) that satisfies additional conditions. In particular, it is a relative cobordism between two manifolds with boundaries. We define energy as the integral of the distance function to the first of these boundary manifolds. Composition of signals is composition of cobordisms. A "time variable" can appear explicitly if it is explictly given (for example, if the manifold is of the form ). Otherwise, there is no designated "time dimension", although the cobordism may implicitly indicate the presence of dynamics. We interpret a local deformation of the metric as noise. The assumptions on M allow to define a map that we call a Fourier transform. We prove inequalities that illustrate the properties of energy of signals in this setting.
Paper Structure (6 sections, 2 theorems, 54 equations, 6 figures)

This paper contains 6 sections, 2 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Cobordisms as signals
  3. Proofs
  4. Proof of Theorem \ref{['thmain1']}.
  5. Proof of Theorem \ref{['thmain2']}.
  6. Conclusions

Key Result

Theorem 2.6

Let $M=(M;X,Y,\Sigma=A\sqcup B;\partial X, \partial Y)$ be a signal. $(i)$ Then $(ii)$ Let $(U, h)$ be noise, where for some $p\in M$ and $\delta>0$. Let $0<\delta_0<\delta$ and let $0<\varepsilon<1$. Then there is a smooth function $a_{\varepsilon}:M\to\mathbb R$ such that and as $\varepsilon\to 0$ where $(iii)$ Let $(U,h)$ be noise as in $(ii)$ and let $M'$ be a filter such that $X_1'=X_1$,

Figures (6)

  • Figure 1: The "empty squares" are the zero entries of the rectangular matrix.
  • Figure 2: A soundform of the word "dot".
  • Figure 3: A cobordism $(X;X_1,X_2)$.
  • Figure 4: A relative cobordism $(M;X,Y,\Sigma=A\cup B;\partial X, \partial Y)$, $k=0$.
  • Figure 5: A cobordism $(X;X_1,X_2)$, $n_1=2$, $n_2=3$.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8