The Geometry of Manifold Learning: Singular foliations via Deep Learning models

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.

Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary, balanced by the von Neumann entropy of the network's weight space. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.

Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications. One recent focus is the generalization of Euclidean fully connected (FC) and convolutional layers to non-Euclidean geometries. However, previous approaches typically focus on a few selected manifolds and rely on specific properties of the target manifold. In contrast, this work proposes a framework for constructing FC and convolutional layers over computationally tractable Riemannian spaces. This framework incorporates several previous FC layers across different geometries as special cases and is instantiated on ten representative manifolds, including three hyperbolic models, five geometries of the symmetric positive definite (SPD) manifold, and two Grassmannian perspectives. Experiments on different manifolds demonstrate the effectiveness and applicability of our approach. Code can be found at https://github.com/GitZH-Chen/RieTrans.