Signals as submanifolds, and configurations of points

TL;DR

This paper develops an abstract geometric framework for signal propagation using submanifolds in Riemannian manifolds, deriving energy bounds for various signal spaces including Gaussian parameters and point configurations.

Contribution

It introduces a novel geometric approach to modeling signals as submanifolds and derives energy inequalities for different signal configuration spaces.

Findings

Derived upper and lower energy bounds for signals modeled as submanifolds.

Established energy inequalities for Gaussian parameter spaces.

Analyzed the role of time and graph embeddings in signal energy bounds.

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.