  We propose a method for learning topology-preserving data representations
(dimensionality reduction). The method aims to provide topological similarity
between the data manifold and its latent representation via enforcing the
similarity in topological 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.
