Aggregation sheaves for greedy modal decompositions

Abstract

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 linearly on the weights within the superposition and nonlinearly upon several other parameters. The particular focus of this article is upon finding both the weights and the parameters when the number of modes is not known in advance. 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. It is shown that minimizing the local consistency radius within the aggregation sheaf is guaranteed to solve all modal decomposition problems. Since the modes may or may not be well-separated, a greedy algorithm that identifies the most distinct modes first may not work reliably.