Spectral Echolocation via the Wave Embedding

TL;DR

This paper introduces a novel spectral embedding method inspired by wave behavior, enhancing data representation by simulating low-frequency waves and refining metrics for better dimensionality reduction.

Contribution

It proposes a new pre-processing step using wave simulation on spectral embeddings, improving the quality of data embeddings for various PDEs including the heat equation.

Findings

Effective in practice for spectral embedding enhancement

Works with multiple partial differential equations

Yields improved dimensionality reduction results

Abstract

Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position as well as change in time of the wave to obtain a refined metric to which classical methods of dimensionality reduction can then applied. This is motivated by the behavior of waves, symmetries of the wave equation and the hunting technique of bats. It is shown to be effective in practice and also works for other partial differential equations -- the method yields improved results even for the classical heat equation.