Matrix Manifold Neural Networks++
Résumé
Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing
interest in various applied areas. For instance, DNNs on spherical and hyperbolic
manifolds have been designed to solve a wide range of computer vision and nature
language processing tasks. One of the key factors that contribute to the success of
these networks is that spherical and hyperbolic manifolds have the rich algebraic
structures of gyrogroups and gyrovector spaces. This enables principled and effective
generalizations of the most successful DNNs to these manifolds. Recently,
some works have shown that many concepts in the theory of gyrogroups and gyrovector
spaces can also be generalized to matrix manifolds such as Symmetric
Positive Definite (SPD) and Grassmann manifolds. As a result, some building
blocks for SPD and Grassmann neural networks, e.g., isometric models and multinomial
logistic regression (MLR) can be derived in a way that is fully analogous
to their spherical and hyperbolic counterparts. Building upon these works, we
design fully-connected (FC) and convolutional layers for SPD neural networks.
We also develop MLR on Symmetric Positive Semi-definite (SPSD) manifolds,
and propose a method for performing backpropagation with the Grassmann logarithmic
map in the projector perspective. We demonstrate the effectiveness of the
proposed approach in the human action recognition and node classification tasks.