Sketching the Heat Kernel: Using Gaussian Processes to Embed Data

• URL: http://arxiv.org/abs/2403.07929v1
• Date: Fri, 1 Mar 2024 22:56:19 GMT
• Title: Sketching the Heat Kernel: Using Gaussian Processes to Embed Data
• Authors: Anna C. Gilbert, Kevin O'Neill,
• Abstract summary: We introduce a novel, non-deterministic method for embedding data in low-dimensional Euclidean space based on realizations of a Gaussian process depending on the geometry of the data. Our method demonstrates further advantage in its robustness to outliers.
• Score: 4.220336689294244
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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