Aggregation sheaves for greedy modal decompositions

TL;DR

This paper introduces aggregation sheaves, a sheaf-theoretic framework that guarantees solutions to nonlinear modal decomposition problems, especially when the number of modes is unknown, and analyzes the behavior of greedy algorithms in this context.

Contribution

It develops a new sheaf-theoretic formalism called aggregation sheaves to analyze and solve nonlinear modal decomposition problems with unknown mode count.

Findings

Minimizing local consistency radius solves modal decomposition.

Aggregation sheaves characterize greedy algorithm behavior.

The formalism handles modes that are not well-separated.

Abstract

This article develops a new theoretical basis for decomposing signals that are formed by the linear superposition of a finite number of modes. Each mode depends nonlinearly upon several parameters; we seek both these parameters and the weights within the superposition. The particular focus of this article is upon solving the decomposition problem when the number of modes is not known in advance. A sheaf-theoretic formalism is introduced that describes all modal decomposition problems, and it is shown that minimizing the local consistency radius within this sheaf is guaranteed to solve them. Since the modes may or may not be well-separated, a greedy algorithm that identifies the most distinct modes first may not work reliably. This article introduces a novel mathematical formalism, aggregation sheaves, and shows how they characterize the behavior of greedy algorithms that attempt to solve modal decomposition problems.