Sketching the Heat Kernel: Using Gaussian Processes to Embed Data
<|reference_start|>Sketching the Heat Kernel: Using Gaussian Processes to Embed Data: 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.<|reference_end|>
@article{gilbert2024sketching,
    title={Sketching the Heat Kernel: Using Gaussian Processes to Embed Data},
    author={Anna C. Gilbert, Kevin O'Neill},
    journal={arXiv preprint arXiv:2403.07929},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.07929},
    primaryClass={cs.LG cs.NA math.NA stat.ML}
}