Learning Geometrically and Topologically Aligned Representations with Topologically Regularized Autoencoders

The manifold hypothesis states that real-world data often lies close to a low-dimensional manifold. This implies that "data has a shape" and exhibits topological features such as holes and voids, but also geometrical features such as curvature. Manifold learning aims to exploit this to find meaningful low-dimensional representations and visualizations of high-dimensional data to gain further insights and use them for downstream tasks. In this talk, I discuss some fundamental ideas of representation learning and present preliminary results from an empirical study on how autoencoders with a persistent homology-based topological regularization term could be used to learn latent representations of data that are not only topologically aligned but also preserve the extrinsic curvature of the data.