Sketching the Heat Kernel: Using Gaussian Processes to Embed Data
This provides a novel embedding technique for data analysis, though it builds on prior theoretical work and appears incremental in its application.
The paper tackles the problem of embedding data in low-dimensional Euclidean space by introducing a non-deterministic method based on Gaussian processes with a heat kernel covariance, which approximates diffusion distances probabilistically and shows robustness to outliers.
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-Loève 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.