On the Impact of Topological Regularization on Geometrical and Topological Alignment in Autoencoders: An Empirical Study

Samuel Graepler, Nico Scherf, Anna Wienhard, Diaaeldin Taha

Keywords: Representation Learning, Manifold Learning, Geometric Deep Learning, Autoencoder

TL;DR: We empirically investigate the impact of topological regularization on the topological and geometrical alignment between data and latent representations in autoencoders, using extrinsic curvature estimation and synthetic manifolds.

Abstract: We present a comparative empirical study on the impact of topological regularization on autoencoders (AEs) and variational autoencoders (VAEs) across six synthetic datasets with known topology and curvature. Particularly, we probe the alignment of the topology and geometry of the dimensionality-reduced latent representation with that of the data. To quantify geometrical alignment, we estimate the mean extrinsic curvature of the latent embedding by fitting local quadrics. We find that topological regularization can significantly improve the geometrical alignment of latent and data, even when the training objective emphasizes topological alignment alone, without regard for reconstruction quality

Submission Number: 48