Information geometry of the Otto metric

In this talk, I present a novel extension of information geometry to the setting of Wasserstein geometry and highlight its applications in machine learning. The central structure of information geometry consists of a Riemannian manifold together with a pair of dual affine connections. Classically, this structure arises from a statistical model endowed with the Fisher–Rao metric, the mixture connection, and its dual, the exponential connection. While this framework captures fundamental statistical properties, it does not account for the metric structure of the underlying sample space.

Motivated by this limitation, I propose an extension to the Wasserstein setting. Within this framework, I introduce the dual of the mixture connection with respect to the Otto metric, yielding a novel form of exponential connection. This leads to a new dual structure comprising the mixture connection, the Otto metric as the Riemannian metric, and the newly defined exponential connection.

I derive the geodesic equation associated with this exponential connection and show that it coincides with the Kolmogorov forward equation of a gradient flow, also known as the continuity equation in Wasserstein geometry. Furthermore, I construct the canonical contrast function of the proposed dual structure, which I term the Wasserstein–Kullback–Leibler divergence, and demonstrate how it addresses limitations of the classical Kullback–Leibler divergence.