Identifying latent distances with Finslerian geometry

Riemannian geometry provides powerful tools to explore the latent space of generative models while preserving the inherent structure of the data manifold. Lengths, energies and volume measures can be derived from a pullback metric, defined through the immersion that maps the latent space to the data space. With this in mind, most generative models are stochastic, and so is the pullback metric. Manipulating stochastic objects is strenuous in practice. In order to perform operations such as interpolations, or measuring the distance between data points, we need a deterministic approximation of the pullback metric. In this work, we are defining a new metric as the expected length derived from the stochastic pullback metric. We show this metric is Finslerian, and we compare it with the expected pullback metric. In high dimensions, we show that the metrics converge to each other at a rate of $\mathcal{O}\left(\frac{1}{D}\right)$.