Manifold Topology Divergence: a Framework for Evaluating Knowledge Manifolds
Authors: Serguei Barannikov, Ilya Trofimov, Grigorii Sotnikov, Ekaterina Trimbach, Alexander Korotin, Alexander Filippov, Evgeny Burnaev
Summary: We develop a framework for evaluating knowledge manifolds, aimed, specifically, in direction of the analysis of deep generative fashions. We describe a novel device, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional house, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based mostly on the Cross-Barcode, we introduce the Manifold Topology Divergence rating (MTop-Divergence) and apply it to evaluate the efficiency of deep generative fashions in numerous domains: photos, 3D-shapes, time-series, and on completely different datasets: MNIST, Trend MNIST, SVHN, CIFAR10, FFHQ, chest X-ray photos, market inventory knowledge, ShapeNet. We show that the MTop-Divergence precisely detects numerous levels of mode-dropping, intra-mode collapse, mode invention, and picture disturbance. Our algorithm scales nicely (primarily linearly) with the rise of the dimension of the ambient high-dimensional house. It is likely one of the first TDA-based sensible methodologies that may be utilized universally to datasets of various sizes and dimensions, together with those on which the newest GANs within the visible area are skilled. The proposed methodology is area agnostic and doesn’t depend on pre-trained networks.
