Geometric Representation Learning on Riemannian Manifolds for Multidimensional Signal Decoding and Modeling

Riemannian deep learning studies how to design and train neural models when data representations and internal states lie on non-Euclidean manifolds rather than in vector spaces. This thesis develops geometry-consistent learning methods on two manifolds that frequently arise in practice: the manifold of symmetric positive definite (SPD) matrices and the Stiefel manifold of orthonormal frames. The work is motivated by the fact that covariance descriptors and related second-order statistics encode invariances and constraints that standard Euclidean pipelines do not preserve. Treating such objects as ordinary vectors can distort distances and averages, exacerbate numerical pathologies under ill-conditioning, and reduce robustness under distribution shifts. These challenges are especially evident in electroencephalogram decoding for brain--computer interfaces, where discriminative information is often carried by covariance structure that varies across time, sessions, and individuals. The thesis introduces scalable architectures that preserve manifold validity throughout training. A central contribution is a learnable orthogonal basis transformation, optimized under Stiefel constraints and applied via symmetry-preserving operations on covariance representations, enabling geometry-consistent alignment and fusion of multiple within-trial covariance views before log-domain processing. The thesis further develops graph-structured manifold learning that couples node-wise SPD representations with geometry-aware transport and message passing, providing relational modeling while controlling computational cost through structured, rank-reduced operators. Beyond supervised decoding, the thesis advances representation learning and generation on manifolds to address label scarcity and limited coverage of training distributions. It proposes self-supervised objectives tailored to covariance representations, yielding label-efficient, shift-robust features; introduces structure-preserving conditional generative modeling with Riemannian regularization for manifold-consistent augmentation; and develops multiscale diffusion-style modeling of manifold-valued dynamics using stochastic processes that preserve symmetry and positive definiteness. Together, these contributions provide a coherent set of principles, architectures, and generative models for geometry-consistent learning on SPD and Stiefel manifolds, with practical guidance for building stable and generalizable electroencephalogram (EEG) decoding systems under distribution shift.