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 data manifolds, aimed, notably, in route of the analysis of deep generative fashions. We describe a novel machine, 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 completely on the Cross-Barcode, we introduce the Manifold Topology Divergence rating (MTop-Divergence) and apply it to guage the effectivity of deep generative fashions in fairly just a few domains: photos, 3D-shapes, time-series, and on absolutely fully totally different datasets: MNIST, Growth MNIST, SVHN, CIFAR10, FFHQ, chest X-ray photos, market inventory data, ShapeNet. We current that the MTop-Divergence precisely detects fairly just a few ranges of mode-dropping, intra-mode collapse, mode invention, and film disturbance. Our algorithm scales correctly (primarily linearly) with the rise of the dimension of the ambient high-dimensional house. It is likely considered one of many first TDA-based smart methodologies which can be utilized universally to datasets of assorted sizes and dimensions, together with these on which the latest GANs all through the seen area are skilled. The proposed methodology is area agnostic and wouldn’t rely upon pre-trained networks.
