Deep Riemannian Manifold Learning

Differential Geometry for ML Workshop at NeurIPS

Abstract

We present a new class of learnable Riemannian manifolds with a metric parameterized by a deep neural network. The core manifold operations–specifically the Riemannian exponential and logarithmic maps–are solved using approximate numerical techniques. Input and parameter gradients are computed with an adjoint sensitivity analysis. This enables us to fit geodesics and distances with gradient-based optimization of both on-manifold values and the manifold itself. We demonstrate our method’s capability to model smooth, flexible metric structures in graph embedding tasks.

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