Riemannian Transformation Layers for General Geometries
Keywords: Manifold Learning, Representation Learning, Riemannian Manifolds
TL;DR: We generalize the transformation layers, such as fully connected and convolutional layer, into Riemannian geometries, and manifest our framework on the different geometreis over the SPD and Grassmannian manifolds.
Abstract: Recently, deep neural networks on manifold-valued representations have garnered significant attention in various machine learning applications. Several studies have attempted to generalize traditional Euclidean transformation layers, such as Fully Connected (FC) and convolutional layers, to non-Euclidean geometries. However, the previous approaches typically focus on a select few manifolds and rely on the specific properties of the target manifold. In this work, we propose a theoretical framework for constructing Riemannian FC and convolutional layers over general geometries, providing broader applicability. Utilizing this framework, we design convolutional networks across five distinct geometries of the Symmetric Positive Definite (SPD) manifold, as well as networks under two Grassmannian perspectives. Extensive experiments demonstrate that the proposed Riemannian convolutional networks significantly outperform existing SPD and Grassmannian networks.
Primary Area: learning on graphs and other geometries & topologies
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Submission Number: 469
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