Fast and Robust Shortest Paths on Manifolds Learned from Data

A longstanding goal in machine learning is to build models that are invariant to irrelevant transformations of the data, as this can remove factors that are otherwise arbitrarily determined. For instance, in nonlinear latent variable models, the latent variables are generally unidentifiable as the latent space is by design not invariant to reparametrizations. Enforcing a Riemannian metric in the latent space that is invariant to reparametrizations alleviate this identifiability issue,which significantly boosts model performance and interpretability [Arvanitidis et al., 2018, Tosi et al., 2014]. Irrelevant transformations of the data can alternatively be factored out by only modeling local behavior of the data; geometrically this can be viewed as having a locally adaptive inner product 1 1.4 1.2 1 0.8 0.6 0.4 0.2 0