Learning Complex Geometric Structures from Data with Deep Riemannian Manifolds

Aaron Lou, Maximilian Nickel, Mustafa Mukadam, Brandon Amos

Keywords: manifold, geometry, graph embedding, geodesic, differential equations, BVP, differentiable programming

Abstract: We present Deep Riemannian Manifolds, a new class of neural network parameterized Riemannian manifolds that can represent and learn complex geometric structures. To do this, we first construct a neural network which outputs symmetric positive definite matrices and show that the induced metric can universally approximate all geometries. We then develop differentiable solvers for core manifold operations like the Riemannian exponential and logarithmic map, allowing us to train the manifold parameters in an end-to-end machine learning system. We apply our method to learn 1) low-distortion manifold graph embeddings and 2) the underlying manifold of geodesic data. In addition to improving upon the baselines, our ability to directly optimize the Riemannian manifold brings to light new perspectives with which to view these tasks.

One-sentence Summary: We construct neural network parameterized Riemannian manifolds and differentiable numerical manifold operations to fit the geometry of data.