Geometry, Topology and Discrete Symmetries Revealed by Deep Neural Networks

Explore the intersection of universality efforts and manifold learning in this 36-minute lecture by Maarten de Hoop from Rice University. Delve into the question of which neural network architectures can universally approximate maps between topologically interesting manifolds. Examine the low-dimensional manifold hypothesis underlying inverse problems and its implications for data analysis. Discover a novel network architecture combining injective flows and coordinate projections that universally approximates local diffeomorphisms between compact smooth submanifolds in Euclidean spaces. Learn how this network enables multi-valued inversion and handles cases where the target map changes topology. Investigate applications in supervised learning for recovering group actions of finite group invariant maps and in unsupervised learning for informing topologically expressive starting spaces in generative models.