Signals as submanifolds, and configurations of points

2024-08-27

arXiv.org

AI Summary

This work addresses the problem of energy estimation for signal propagation on submanifolds of Riemannian manifolds. Methodologically, it introduces a geometric signal modeling paradigm wherein signals are defined intrinsically as submanifolds—not scalar-valued functions—thereby unifying signal theory with differential geometry. The approach integrates submanifold theory, energy functional analysis, configuration space topology, and graph embedding techniques to derive, for the first time, explicit upper and lower bounds on signal propagation energy within canonical geometric parameter spaces—including Gaussian distribution manifolds and point configuration spaces. It further systematically characterizes how temporal evolution and graph embeddings affect energy constraints. The primary contribution is the establishment of the first Riemannian-geometric framework for energy analysis of submanifold-valued signals, providing both a rigorous theoretical foundation and computationally tractable quantification tools for geometric signal processing.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

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.

Problem

• Studying signals as submanifolds in Riemannian manifolds
• Deriving energy bounds for Gaussian parameter spaces
• Analyzing time and graph embeddings in signal propagation

Innovation

• Signals modeled as Riemannian submanifolds
• Energy bounds for Gaussian parameter spaces
• Graph embeddings incorporate time dynamics