The paper introduces Cartan flow matching, a framework for training flow matching models on Riemannian symmetric spaces such as spheres, hyperbolic space, and Grassmannians. By leveraging the algebraic structure of these manifolds, the authors reformulate flow matching on a subspace of the Lie algebra of the isometry group, thereby linearizing the problem and eliminating the need for geodesic interpolation paths. The framework is demonstrated on real Grassmannians SO(n)/SO(k) × SO(n-k).
arXiv:2609.35436v2 Announce Type: replace Abstract: Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications....
arXiv:2607. 19305v2 Announce Type: replace-cross Abstract: 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.
arXiv:2607. 19305v1 Announce Type: cross Abstract: 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.
arXiv:2607. 08987v1 Announce Type: new Abstract: Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures.
arXiv:2607. 00556v1 Announce Type: cross Abstract: While recent advancements like the Poincar\'e ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold.
While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals.
Geometric foundation models, such as the Visual Geometry Grounded Transformer (VGGT), provide strong 3D priors from unposed images. However, such models operate purely in a feed-forward, deterministic regime, \ie~they cannot generate plausible geometry beyond what the input views directly support.
arXiv:2607. 08783v1 Announce Type: cross Abstract: Manifold-valued measurements are prevalent in various machine learning tasks.
arXiv:2512. 20043v3 Announce Type: replace Abstract: Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning.
arXiv:2606. 01847v1 Announce Type: cross Abstract: Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the $\textbf{Euclidean Fallacy}$: representing SE(3) poses as flat $\mathbb{R}^{12}$ vectors.
arXiv:2510. 09468v3 Announce Type: replace Abstract: Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective.
The paper introduces a novel generalization of the Adam optimizer to manifold settings, specifically targeting homogeneous spaces such as the Stiefel, symplectic Stiefel, and Grassmann manifolds. By exploiting a global tangent space representation (the Lie subspace), the authors eliminate the need for projection steps and enable all Adam operations to be performed directly on these manifolds. The new optimizer is applied to train transformers and a symplectic autoencoder, achieving orthogonality constraints to machine precision and outperforming existing methods.