Algebraic SPD and Correlation Geometry: A Gyro Approach
Keywords: Correlation matrices, SPD manifolds, Gyrovector spaces
TL;DR: We propose the gyro-structure for full-rank correlation matrices and SPD manifolds under power-Euclidean geometry.
Abstract: The generalization of Deep Neural Networks (DNNs) to Riemannian manifolds has garnered significant attention across various scientific fields. Recent studies have demonstrated that several manifolds, including hyperbolic, spherical, Symmetric Positive Definite (SPD), and Grassmann manifolds, admit gyro-structures—powerful algebraic structures that enable the principled extension of DNNs to manifolds. Inspired by these advancements, we introduce a novel gyro-structure for SPD manifolds, leveraging the flexible and powerful Power-Euclidean (PE) geometry. Moreover, full-rank correlation matrices, which are scale-invariant, serve as compact representations of SPD manifolds. Consequently, we propose two novel gyro-structures for correlation matrix manifolds, based on two theoretically and empirically convenient metrics: Euclidean-Cholesky (EC) and log-Euclidean-Cholesky (LEC) geometries. Extensive experiments on knowledge graph completion tasks validate the effectiveness of our proposed gyro-structures.
Primary Area: learning on graphs and other geometries & topologies
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Submission Number: 515
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