🤖 AI Summary
Existing manifold-based deep learning approaches are often limited to specific manifolds, rely on Euclidean approximations, or involve computationally expensive and numerically unstable geometric operations. This work proposes a unified Riemannian deep learning framework whose core innovations include generalizing batch normalization to Lie and rotation groups, extending multinomial logistic regression to arbitrary Riemannian manifolds, and designing an adaptive, efficient metric for symmetric positive definite (SPD) matrices. The framework integrates generalized batch normalization, Riemannian multinomial regression, unconstrained modeling in hyperbolic space, Busemann function learning, and Cholesky/Log-Euclidean geometries. Theoretical analysis and experiments demonstrate that the proposed method achieves superior performance and computational efficiency across diverse tasks in computer vision, signal processing, graph learning, and genomics.
📝 Abstract
Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.