  Riemannian geometry provides us with powerful tools to explore the latent
space of generative models while preserving the underlying structure of the
data. The latent space can be equipped it with a Riemannian metric, pulled back
from the data manifold. With this metric, we can systematically navigate the
space relying on geodesics defined as the shortest curves between two points.
Generative models are often stochastic, causing the data space, the Riemannian
metric, and the geodesics, to be stochastic as well. Stochastic objects are at
best impractical, and at worst impossible, to manipulate. A common solution is
to approximate the stochastic pullback metric by its expectation. But the
geodesics derived from this expected Riemannian metric do not correspond to the
expected length-minimising curves. In this work, we propose another metric
whose geodesics explicitly minimise the expected length of the pullback metric.
We show this metric defines a Finsler metric, and we compare it with the
expected Riemannian metric. In high dimensions, we prove that both metrics
converge to each other at a rate of $O\left(\frac{1}{D}\right)$. This
convergence implies that the established expected Riemannian metric is an
accurate approximation of the theoretically more grounded Finsler metric. This
provides justification for using the expected Riemannian metric for practical
implementations.
