Identifying latent distances with Finslerian geometry

Riemannian geometry has been shown useful to explore the latent space of models of high dimensional data. This latent space is learnt via a stochastic smooth mapping, and a deterministic approximation of the metric is required. Yet, this approximation is ad-hoc and doesn't lead to interpretable quantities, such as the curve length. Here, we are defining a new metric as the expectation of the stochastic length induced by this smooth mapping. We show that this norm is a Finsler metric. We compare this Finsler metric with the previously studied expected Riemannian metric, and we show that in high dimensions, these metrics converge to each other.