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

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