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.

This work addresses the finite-time analysis of nonsmooth nonconvex stochastic optimization under Riemannian manifold constraints. We adapt the notion of Goldstein stationarity to the Riemannian setting as a performance metric for nonsmooth optimization on manifolds. We then propose a Riemannian Online to NonConvex (RO2NC) algorithm, for which we establish the sample complexity of $O(ε^{-3}δ^{-1})$ in finding $(δ,ε)$-stationary points. This result is the first-ever finite-time guarantee for fully nonsmooth, nonconvex optimization on manifolds and matches the optimal complexity in the Euclidean setting. When gradient information is unavailable, we develop a zeroth order version of RO2NC algorithm (ZO-RO2NC), for which we establish the same sample complexity. The numerical results support the theory and demonstrate the practical effectiveness of the algorithms.