A Riemannian View on Active Subspaces

We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing methods for generative modeling on manifolds rely on substantial geometric information such as geodesic curves or eigenfunctions of the Laplace-Beltrami operator and, as a result, they are limited to manifolds where such information is available. In contrast, our method, built on a projection scheme, can be applied to more general manifolds, as it only requires being able to evaluate the value and the first order derivatives of the function that defines the submanifold. We provide a theoretical analysis of our method in the continuous-time limit, which elucidates the connection between our RDDPMs and score-based generative models on manifolds. The capability of our method is demonstrated on datasets from previous studies and on new datasets sampled from two high-dimensional manifolds, i.e. $\mathrm{SO}(10)$ and the configuration space of molecular system alanine dipeptide with fixed dihedral angle.

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.

Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.