Manifold Learning with Sparse Regularised Optimal Transport
This work addresses the challenge of detecting latent manifolds in noisy datasets for applications like cell, document, image, and molecule analysis, representing an incremental improvement with a novel computational scheme.
The authors tackled the problem of manifold learning from noisy, high-dimensional data by proposing a method using symmetric optimal transport with quadratic regularization to create a sparse affinity matrix, and demonstrated that it outperforms competing methods in numerical experiments.
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.