Only Bayes should learn a manifold (on the estimation of differential geometric structure from data)
This addresses a foundational issue in machine learning for researchers working on manifold learning and generative models, highlighting limitations in current methods.
The paper tackles the problem of learning the differential geometric structure of data manifolds, showing that non-probabilistic methods fail to recover this structure under typical regularizations, while probabilistic methods can succeed under reasonable priors, though fully exploiting it requires new stochastic extensions to Riemannian geometry.
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostly linear manifolds or spaces equipped with teleports. To properly learn the differential geometric structure, non-probabilistic methods must apply regularizations that enforce large gradients, which go against common wisdom. We repeat the analysis for probabilistic methods and find that under reasonable priors, the geometric structure can be recovered. Fully exploiting the recovered structure, however, requires the development of stochastic extensions to classic Riemannian geometry. We take early steps in that regard. Finally, we partly extend the analysis to modern models based on neural networks, thereby highlighting geometric and probabilistic shortcomings of current deep generative models.