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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.07929 |
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| _version_ | 1866916157365157888 |
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| author | Gilbert, Anna C. O'Neill, Kevin |
| author_facet | Gilbert, Anna C. O'Neill, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sketching the Heat Kernel: Using Gaussian Processes to Embed Data Gilbert, Anna C. O'Neill, Kevin Machine Learning Numerical Analysis 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. |
| title | Sketching the Heat Kernel: Using Gaussian Processes to Embed Data |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2403.07929 |