Riemannian Manifolds
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Related Articles from SNS
Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds
Announce Type: cross Abstract: Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
Post-processed frozen-flow methods for the long time sampling of ergodic dynamics on Riemannian manifolds
Announce Type: new Abstract: In this work, we propose a novel intrinsic approach to the approximation of ergodic SDEs on Riemannian manifolds, which include Riemannian Langevin dynamics. In opposition to the standard extrinsic approaches such as penalization methods and projection methods, our methodology does not use embeddings or coordinates and only relies on natural geometric operations: geodesics, parallel transport,... We give a criterion for high order of accuracy for the invariant...
Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle
arXiv:2605.17502v2 Announce Type: replace-cross Abstract: On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids...
Learning from Demonstrations over Riemannian Manifolds using Neural ODEs: An Extended Abstract
Announce Type: new Abstract: Learning from demonstratins (LfD) is usually performed over Euclidean spaces, while the robot state, e.g. orientation, naturally evolves over curved spaces. Therefore, to ensure natural, complex motion generation, we investigate learning from demonstrations over Riemannian manifolds that are capable of encoding both position and orientation data.
Inversion-Free Natural Gradient Descent on Riemannian Manifolds
arXiv:2604.02969v2 Announce Type: replace-cross Abstract: The natural gradient method is a central tool for statistical optimisation, but its broader application is hindered by the assumption of a Euclidean parameter space, the repeated estimation of the Fisher information matrix (FIM), and the computational cost of its subsequent inversion. This paper proposes an intrinsic, inversion-free natural gradient method for statistical models whose parameters lie on general Riemannian manifolds....
Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering
arXiv:2605.24942v2 Announce Type: replace Abstract: Steering a language model - intervening on its internal activations to change downstream behaviour - has recently expanded beyond linear interpolation to nonlinear methods such as angular and kernelized steering, which define intervention transformations without learning an explicit geometry over paths in activation space. Freshly introduced geometry-aware manifold methods do learn such a geometry, but require labelled class centroids...
Geodesic Flow Matching on a Riemannian Degradation Manifold for Blind Image Restoration
Announce Type: new Abstract: Blind image restoration requires recovering clean images from observations corrupted by unknown and potentially mixed degradations. While recent deterministic flow-based methods model restoration as transport processes that map degraded images to clean ones, they typically rely on Euclidean interpolation, implicitly assuming linear degradation geometry. In this paper, we explicitly model degradations as points on a low-dimensional Riemannian manifold and...
Knowledge Manifold: A Riemannian Geometric Framework for Semantic Mapping and Geodesic Analysis of Scientific Literature
arXiv:2606.05907v1 Announce Type: new Abstract: We present the knowledge manifold: a Riemannian geometric space in which a corpus of documents is arranged according to semantic positional relationships derived from character n-gram TF-IDF representations. The framework proceeds in five tightly coupled stages. First, each document is converted to a character-level n-gram TF-IDF vector (4-7 grams, up to 250,000 features, L2-normalized) and embedded in a two-dimensional knowledge map via...
Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds
arXiv:2606.02879v1 Announce Type: new Abstract: Informed sampling techniques accelerate sampling-based motion planners by focusing the search on promising regions of the state space, yet most existing methods rely on Euclidean heuristics that become inadmissible under configuration-dependent Riemannian metrics. While scalar eigenvalue bounds restore admissibility by uniformly scaling the Euclidean distance, they discard the directional structure of the metric, producing overly conservative...
Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
arXiv:2509.07779v2 Announce Type: replace-cross Abstract: We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure.