Sketching the Heat Kernel: Using Gaussian Processes to Embed Data
Anna C. Gilbert Kevin O'Neill
This paper introduces a novel, non-deterministic method for embedding data in
low-dimensional Euclidean space based on computing realizations of a Gaussian
process depending on the geometry of the data. This type of embedding first
appeared in (Adler et al, 2018) as a theoretical model for a generic manifold
in high dimensions.
  In particular, we take the covariance function of the Gaussian process to be
the heat kernel, and computing the embedding amounts to sketching a matrix
representing the heat kernel. The Karhunen-Lo\`eve expansion reveals that the
straight-line distances in the embedding approximate the diffusion distance in
a probabilistic sense, avoiding the need for sharp cutoffs and maintaining some
of the smaller-scale structure.
  Our method demonstrates further advantage in its robustness to outliers. We
justify the approach with both theory and experiments.