On Symmetries and Metrics in Geometric Inference

Abstract: Spaces of data naturally carry intrinsic geometry. Statistics and machine learning can leverage on this rich structure in order to achieve efficiency and semantic generalization. Extracting geometry from data is therefore a fundamental challenge which by itself defines a statistical, computational and unsupervised learning problem. To this end, symmetries and metrics are two fundamental objects which are ubiquitous in continuous and discrete geometry. Both are suitable for data-driven approaches since symmetries arise as interactions and are thus collectable in practice while metrics can be induced locally from the ambient space. In this thesis, we address the question of extracting geometry from data by leveraging on symmetries and metrics. Additionally, we explore methods for statistical inference exploiting the extracted geometric structure. On the metric side, we focus on Voronoi tessellations and Delaunay triangulations, which are classical tools in computational geometry. Based on them, we propose novel non-parametric methods for machine learning and statistics, focusing on theoretical and computational aspects. These methods include an active version of the nearest neighbor regressor as well as two high-dimensional density estimators. All of them possess convergence guarantees due to the adaptiveness of Voronoi cells. On the symmetry side, we focus on representation learning in the context of data acted upon by a group. Specifically, we propose a method for learning equivariant representations which are guaranteed to be isomorphic to the data space, even in the presence of symmetries stabilizing data. We additionally explore applications of such representations in a robotics context, where symmetries correspond to actions performed by an agent. Lastly, we provide a theoretical analysis of invariant neural networks and show how the group-theoretical Fourier transform emerges in their weights. This addresses the problem of symmetry discovery in a self-supervised manner.