Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices

Federico Lopez, Maria Beatrice Pozzetti, Steve J Trettel, Michael Strube, Anna Wienhard

Keywords: symmetric spaces, spd space, spd manifold, symmetric positive definite matrices, spd, riemannian manifold, rotations, reflections, translations, scaling, gyro vector, gyro calculus, gyro groups, gyrocalculus, tangent space optimization, non euclidean optimization, hyperbolic geometry, hyperbolic space, matrix models, non-euclidean geometry, finsler metrics, finsler distance, finsler geometry, vector valued distance, vector valued distance function, riemannian manifold learning, manifold learning, geometric deep learning, graph embeddings, knowledge graph embeddings, item recommendations, question answering

TL;DR: We propose a framework to compute vector-valued distances and adapt Euclidean operations into the SPD manifold

Abstract: We propose the use of the vector-valued distance to compute distances and extract geometric information from the manifold of symmetric positive definite matrices (SPD), and develop gyrovector calculus, constructing analogs of vector space operations in this curved space. We implement these operations and showcase their versatility in the tasks of knowledge graph completion, item recommendation, and question answering. In experiments, the SPD models outperform their equivalents in Euclidean and hyperbolic space. The vector-valued distance allows us to visualize embeddings, showing that the models learn to disentangle representations of positive samples from negative ones.

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Code: https://github.com/fedelopez77/gyrospd