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

NeurIPS 2021spotlight22 citations

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

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.

symmetric spacesspd spacespd manifoldsymmetric positive definite matricesspdriemannian manifoldrotationsreflectionstranslationsscalinggyro vectorgyro calculusgyro groupsgyrocalculustangent space optimizationnon euclidean optimizationhyperbolic geometryhyperbolic spacematrix modelsnon-euclidean geometryfinsler metricsfinsler distancefinsler geometryvector valued distancevector valued distance functionriemannian manifold learningmanifold learninggeometric deep learninggraph embeddingsknowledge graph embeddingsitem recommendationsquestion answering

BibTeX

@inproceedings{
lopez2021vectorvalued,
title={Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices},
author={Federico Lopez and Maria Beatrice Pozzetti and Steve J Trettel and Michael Strube and Anna Wienhard},
booktitle={Advances in Neural Information Processing Systems},
editor={A. Beygelzimer and Y. Dauphin and P. Liang and J. Wortman Vaughan},
year={2021},
url={https://openreview.net/forum?id=9uXILaIam0}
}