This PhD project develops Riemannian deep learning methods at the interface of differential geometry and deep learning. It studies how parsimonious geometric structures, especially Riemannian manifolds, can be used to design data-efficient, robust, and interpretable deep learning algorithms. The project is organized around three connected directions: networks, geometries, and applications. In the networks direction, it generalizes core Euclidean network components, including normalization, transformation, classification, convolution, and attention, to manifold-valued data. A central goal is to construct intrinsic neural network modules that operate directly on curved spaces while remaining compatible with modern deep learning toolchains. In the geometries direction, the project develops networks on expressive yet parsimonious manifolds, including SPD manifolds, correlation manifolds, Grassmannians, and multiple hyperbolic models. It also studies metric and Riemannian geometry as unifying foundations for extending neural network operations across broad classes of non-Euclidean spaces. In the applications direction, the project translates these geometric principles into scalable learning systems for domains where structure, robustness, and interpretability are essential, including EEG decoding, multimodal fusion, and generative modeling. Overall, the project aims to build a coherent theory and practical toolbox for selecting and exploiting suitable geometries in deep learning. Its long-term goal is to expand modern deep learning beyond Euclidean representations toward geometry-aware models that are theoretically principled, computationally practical, and useful for real-world applications.
