AI Seminar: Only Bayes should learn a manifold
Søren Hauberg, Associate Professor at Department of Applied Mathematics and Computer Science, DTU.
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.
This seminar is a part of the AI Seminar Series organised by SCIENCE AI Centre. The series highlights advances and challenges in research within Machine Learning, Data Science, and AI. Like the AI Centre itself, the seminar series has a broad scope, covering both new methodological contributions, ground-breaking applications, and impacts on society.