Spectral Echolocation via the Wave Embedding
This is an incremental improvement for data analysis and dimensionality reduction tasks.
The paper tackles the problem of improving spectral embedding by simulating a low-frequency wave over data to refine the metric for dimensionality reduction, resulting in effective practical performance and improved results even for classical methods like the heat equation.
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.