The Sinkhorn distance: the Riemannian structure of entropic optimal transport

• Applicant: Professor Dr. Bernhard Schmitzer
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 569461875

The formal Riemannian structure induced by optimal transport provides a natural way for comparing and manipulating probability measures and it has become a ubiquitous tool in geometric data analysis. Ongoing challenges are the relatively high algorithmic complexity and the difficulty to estimate optimal transport distances from empirical data in high dimensions. Both issues are remedied by entropic regularization. However, while entropic optimal transport does come with its own rich dynamic structure in the form of Schrödinger bridges, it does not directly induce a metric. This is partially resolved by the de-biased Sinkhorn divergence, but the latter does still not satisfy the triangle inequality, which restricts its applicability, for instance for interpolations, gradient flows, or for local linearization. We have recently introduced a Riemannian structure on the space of probability measures by taking the Hessian of the Sinkhorn divergence as a metric tensor, thus combining the conceptual elegance of the Wasserstein distance with the smoothness of entropic regularization. The resulting distance metrizes weak* convergence and it is equivalent to the norm of the reproducing kernel Hilbert space (RKHS) induced by the entropic transport kernel via a suitable embedding. Under this new distance, translations of measures are geodesics and the distance appears to be robust to fluctuations below the entropic blur scale, making it practical for empirical approximations. Particle methods are currently performing extremely successful on statistical problems such as generative density estimation in high dimensions. In this project we will study the properties of the new distance in more detail and develop its potential as a foundation for geometrically informed particle methods.

• DFG Programme: Research Grants