# Sketching the Heat Kernel: Using Gaussian Processes to Embed Data

arXiv: https://arxiv.org/abs/2403.07929 · HTML: https://mlsanity.com/paper/2403.07929/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 20 references that are in the map.
- First posted on arXiv 1 Mar 2024.

## Builds on (most cited first, 4 of 4)

- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise](https://mlsanity.com/paper/2006.00402/) (2020)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [The embedding dimension of Laplacian eigenfunction maps](https://mlsanity.com/paper/1605.01643/) (2016)

## Abstract (arXiv)

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.