Scaling Optimal Transport to High-Dimensional Gaussian Distributions
Résumé
Although optimal transport (OT) has recently become very popular in machine learning, it faces challenges when dealing with high-dimensional data, such as images or omics data. Current OT approaches for high-dimensional situations rely on projections of the data or measures onto low-dimensional spaces, which inevitably results in information loss. In this work, we consider the case of high-dimensional Gaussian distributions with parsimonious covariance structures and lower intrinsic dimension. We exhibit a simplified closed-form expression of the 2-Wasserstein distance with an efficient and robust calculation procedure based on a lowdimensional decomposition of empirical covariance matrices, without relying on data projections. Furthermore, we provide a closed-form expression for the Monge map, which involves the exact calculation of the square-root and inverse square-root of the source distribution covariance matrix. This approach offers analytical and computational advantages, as demonstrated by our numerical experiments, which quantitatively evaluate these benefits in comparison to existing methods. In addition to being able to compute both the W 2 2 -distance and the transport map, our method outperforms model-free methods, in high dimension, even in the case of non-Gaussian distributions.
Domaines
Statistiques [stat]Origine | Fichiers produits par l'(les) auteur(s) |
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