Manifold Learning with Sparse Regularised Optimal Transport
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917925521195008 |
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| author | Zhang, Stephen Mordant, Gilles Matsumoto, Tetsuya Schiebinger, Geoffrey |
| author_facet | Zhang, Stephen Mordant, Gilles Matsumoto, Tetsuya Schiebinger, Geoffrey |
| contents | Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high dimensional ambient space, however the degrees of freedom intrinsic to the data are usually far fewer than the number of ambient dimensions. The task of detecting a latent manifold along which the data are embedded is a prerequisite for a wide family of downstream analyses. Real-world datasets are subject to noisy observations and sampling, so that distilling information about the underlying manifold is a major challenge. We propose a method for manifold learning that utilises a symmetric version of optimal transport with a quadratic regularisation that constructs a sparse and adaptive affinity matrix, that can be interpreted as a generalisation of the bistochastic kernel normalisation.
We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, establish robustness to heteroskedastic noise and exhibit these results in numerical experiments. We identify a highly efficient computational scheme for computing this optimal transport for discrete data and demonstrate that it outperforms competing methods in a set of examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_09816 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Manifold Learning with Sparse Regularised Optimal Transport Zhang, Stephen Mordant, Gilles Matsumoto, Tetsuya Schiebinger, Geoffrey Machine Learning Statistics Theory 68T01, 62R30 Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high dimensional ambient space, however the degrees of freedom intrinsic to the data are usually far fewer than the number of ambient dimensions. The task of detecting a latent manifold along which the data are embedded is a prerequisite for a wide family of downstream analyses. Real-world datasets are subject to noisy observations and sampling, so that distilling information about the underlying manifold is a major challenge. We propose a method for manifold learning that utilises a symmetric version of optimal transport with a quadratic regularisation that constructs a sparse and adaptive affinity matrix, that can be interpreted as a generalisation of the bistochastic kernel normalisation. We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, establish robustness to heteroskedastic noise and exhibit these results in numerical experiments. We identify a highly efficient computational scheme for computing this optimal transport for discrete data and demonstrate that it outperforms competing methods in a set of examples. |
| title | Manifold Learning with Sparse Regularised Optimal Transport |
| topic | Machine Learning Statistics Theory 68T01, 62R30 |
| url | https://arxiv.org/abs/2307.09816 |