Learning topology-preserving data representations

Abstract

We propose a method for learning topology-preserving data representations (dimen- sionality reduction). The method aims to provide topological similarity between the data manifold and its latent representation via enforcing the similarity in topo- logical features (clusters, loops, 2D voids, etc.) and their localization. The core of the method is the minimization of the Representation Topology Divergence (RTD) between original high-dimensional data and low-dimensional representation in latent space. RTD minimization provides closeness in topological features with strong theoretical guarantees. We develop a scheme for RTD differentiation and apply it as a loss term for the autoencoder. The proposed method “RTD-AE” better preserves the global structure and topology of the data manifold than state-of-the- art competitors as measured by linear correlation, triplet distance ranking accuracy, and Wasserstein distance between persistence barcodes.