Understanding Continuous-depth Networks through the Lens of Homogeneous Ricci Flows
Primary Area: learning on graphs and other geometries & topologies
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Keywords: Riemannian geometry, Time-dependent ODE, Continuous-depth network
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Abstract: The continuous-depth models pioneered by Neural ODEs have sparked a resurgence in exploring dynamic systems based on deep learning prototypes. The studies employed to investigate their theoretical properties mainly rely on Euclidean space, however, the geometric principle of general neural networks has been developed on the Riemannian manifold. Motivated by this open problem, we construct a formalized geometric theory of continuous-depth networks through the lens of homogeneous Ricci flows. From this perspective, the Riemannian metric tensor with coordinate representations learned by the continuous-depth network itself is the closed-form solution of homogeneous Ricci flows. With the presence of Ricci solitons, the Ricci curvature tensor on the underlying data manifold emerges for the first time. This implies that the continuous-depth network governs the Ricci curvature to drive the different kinds of data apart from each other, which is a novel observation between the Ricci curvature and data separation. Toy experiments confirm parts of the proposed theory, as well as provide intuitions and visualizations as to how the Ricci curvature tensor governed by continuous-depth networks evolves on the manifold to operate on data.
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Submission Number: 3025
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