Identifying Latent Distances with Finslerian Geometry

Abstract

Riemannian geometry has been shown useful to explore the latent space of generative models. Effectively, we can endow the latent space with the pullback metric obtained from the data space. Because most generative models are stochastic, this metric will be de facto stochastic, and, as a consequence, a deterministic approximation of the metric is required. Here, we are defining a new metric as the expectation of the stochastic curve lengths induced by the pullback metric. We show this metric is, in fact, a Finsler metric. We compare it with a previously studied expected Riemannian metric, and we show that in high dimensions, the metrics converge to each other.