We study the task of efficiently sampling from a Gibbs distribution dπ = e hdvolg over a Riemannian manifold M via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming his Lipschitz and M has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within ε-Wasserstein distance of π after O(ε 2)steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where hcan be nonconvex and M can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that π satisfies a CD(,) condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by O(ε 2)as well.

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

Attention mechanism has shown great performance and efficiency in a lot of deep learning models, in which relative position encoding plays a crucial role. However, when introducing attention to manifolds, there is no canonical local coordinate system to parameterize neighborhoods.

In all cases, HypGRU achieves the best results when the data are projected to hyperbolic spaces before theyare fed to the network, and all its layers are based on hyperbolic geometry. Results of these networks are obtained using their official code.3,4 We also evaluate a light version of Shift-GCN referred to as Shift-GCN-light, where the numbers of inputand output channels for the input and residual blocks arereduced byafactor of2(thenumber ofinput channels fortheinput block is3). We can also see that whenM = 3, GyroAI-HAUNet outperforms Shift-GCN-light on all the datasets. Overall, whenM = 3, GyroAI-HAUNet is competitive to the best GNN model with far fewer parameters.