Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices
This work addresses geometric modeling challenges for researchers in machine learning and data science, offering incremental improvements in representation learning for structured data.
The authors tackled the problem of computing distances and geometric operations on the manifold of symmetric positive definite matrices by proposing a vector-valued distance and gyrocalculus, resulting in SPD models outperforming Euclidean and hyperbolic equivalents in tasks like knowledge graph completion and question answering.
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.