Towards a General Attention Framework on Gyrovector Spaces for Matrix Manifolds

Deep neural networks operating on non-Euclidean geometries have recently demonstrated impressive performance across various machine-learning applications. Several studies have extended the attention mechanism to different manifolds. However, most existing non-Euclidean attention models are tailored to specific geometries, limiting their applicability. On the other hand, recent studies show that several matrix manifolds, such as Symmetric Positive Definite (SPD), Symmetric Positive Semi-Definite (SPSD), and Grassmannian manifolds, admit gyrovector structures, which extend vector addition and scalar product into manifolds. Leveraging these properties, we propose a Gyro Attention (GyroAtt) framework over general gyrovector spaces, applicable to various matrix geometries.