Manifold Learning with Sparse Regularised Optimal Transport

• URL: http://arxiv.org/abs/2307.09816v2
• Date: Mon, 17 Feb 2025 16:24:09 GMT
• Title: Manifold Learning with Sparse Regularised Optimal Transport
• Authors: Stephen Zhang, Gilles Mordant, Tetsuya Matsumoto, Geoffrey Schiebinger,
• Abstract summary: 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. 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.
• Score: 1.949927790632678
• License:
• Abstract: 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.

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