Identifying latent distances with Finslerian geometry

Alison Pouplin, David Eklund, Carl Henrik Ek, Søren Hauberg

Abstract:

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 $\mathcal{O}\left(\frac{1}{D}\right)$. 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.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=bm2XSzY6o7&noteId=5JJwl7TKAW

Changes Since Last Submission:

The paper presents a Finsler geometry approach for a stochastic modeling problem and compares it to an established Riemannian solution. The Area Chair raised concerns about the lack of clarity regarding why the Finsler approach should be preferred over the Riemannian one.

In response, the abstract and introduction were revised to emphasize that the intention is not to claim that the Finsler metric is better than the Riemannian one. The focus is on addressing a theoretical concern of navigating a stochastic manifold using the expected length-minimizing curves (geodesics), which is possible with the Finsler metric (it is designed this way) but not with the Riemannian metric. The conclusion highlights that since the two metrics are similar when the variance is low or when the number of dimensions increases, the use of the Riemannian metric is further justified in practical cases.

Code: https://github.com/a-pouplin/latent_distances_finsler

Assigned Action Editor: ~Bamdev_Mishra1

License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 1292