  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.
