There is extensive literature on accelerating first-order optimization methods in an Euclidean setting. Under which conditions such acceleration is feasible in Riemannian optimization problems is an active area of research. Motivated by the recent success of silver stepsize methods in the Euclidean setting, we undertake a study of such algorithms in the Riemannian setting. We provide the new class of algorithms determined by the choice of vector transport that allows the silver stepsize acceleration on Riemannian manifolds for the function classes associated with the corresponding vector transport. As a core application, we show that our algorithm recovers the standard Wasserstein gradient descent on the 2-Wasserstein space and, as a result, provides the first provable accelerated gradient method for potential functional optimization problems in the Wasserstein space.

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 3δ 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.

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.

Existing methods for solving Riemannian bilevel optimization (RBO) problems require prior knowledge of the problem's first-and second-order information and curvature parameter of the Riemannian manifold to determine step sizes, which poses practical limitations when these parameters are unknown or computationally infeasible to obtain. In this paper, we introduce the Adaptive Riemannian Hypergradient Descent (AdaRHD) algorithm for solving RBO problems. To our knowledge, AdaRHD is the first method to incorporate a fully adaptive step size strategy that eliminates the need for problem-specific parameters in RBO. We prove that AdaRHD achieves an O(1/ϵ)iteration complexity for finding an ϵ-stationary point, thus matching the complexity of existing non-adaptive methods. Furthermore, we demonstrate that substituting exponential mappings with retraction mappings maintains the same complexity bound. Experiments demonstrate that AdaRHD achieves comparable performance to existing non-adaptive approaches while exhibiting greater robustness.

Consistency models are a class of generative models that enable few-step generation for diffusion and flow matching models. While consistency models have achieved promising results on Euclidean domains like images, their applications to Riemannian manifolds remain challenging due to the curved geometry. In this work, we propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling while respecting the intrinsic manifold constraint imposed by the Riemannian geometry. Leveraging the covariant derivative and exponential-map-based parameterization, we derive the closed-form solutions for both discrete-and continuous-time training objectives for RCM. We then demonstrate theoretical equivalence between the two variants of RCM: Riemannian consistency distillation (RCD) that relies on a teacher model to approximate the marginal vector field, and Riemannian consistency training (RCT) that utilizes the conditional vector field for training. We further propose a simplified training objective that eliminates the need for the complicated differential calculation. Finally, we provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles.

Principal component analysis (PCA) is a major statistical analysis and machine learning tool for dimensional reduction and visualization of high-dimensional datasets [1]. Classical PCA in the Euclidean space is to find the eigenvectors associated with the top eigenvalues of the covariance matrix. Geometrically, PCA can be interpreted as finding the orthogonal directions that maximize the projected data variance to the linear subspace spanned by those directions. Recently, efforts for extending the Euclidean PCA to capture variations for a collection of probability measures have been made [2, 3, 4]. Since the Wasserstein space is an infinite-dimensional curved space, one challenge is to define a proper notion of principal mode of variations in the space of probability measures. In this paper, we take a variational and dynamical perspective of the Euclidean PCA that has robust generalization to the Wasserstein geometry. Specifically, given input data points x1,...,xn in the Euclidean space Rm, performing the standard PCA to find the first principal mode of variation gt = xn +tv passing through the mean xn = n 1 Pn i=1 xi can be reformulated as minimizing the residuals by projecting each data point in the direction v: ˆv1 = argmin

A fundamental challenge in cognitive neuroscience is understanding how cognition emerges from the interplay between structural connectivity (SC) and functional connectivity (FC). Current machine learning approaches typically seek to establish direct mappings from SC to FC associated with specific cognitive states. However, these methods often treat SC and FC as distinct endpoints, failing to capture the coupling relationship throughout the progressive transformation between them. To address this limitation, we propose BrainFlow, a reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribution of FCs from different cognitive tasks.

Class imbalance in graph data presents a significant challenge for effective node classification, particularly in semi-supervised scenarios. In this work, we formally introduce the concept of geometric imbalance, which captures how message passing on class-imbalanced graphs leads to geometric ambiguity among minority-class nodes in the riemannian manifold embedding space. We provide a rigorous theoretical analysis of geometric imbalance on the riemannian manifold and propose a unified framework that explicitly mitigates it through pseudo-label alignment, node reordering, and ambiguity filtering. Extensive experiments on diverse benchmarks show that our approach consistently outperforms existing methods, especially under severe class imbalance. Our findings offer new theoretical insights and practical tools for robust semi-supervised node classification.

Finding low-rank matrix weights is a key technique for addressing the high memory usage and computational demands of large models. Most existing algorithms rely on the factorization of the low-rank matrix weights, which is non-unique and redundant. Their convergence is slow especially when the target low-rank matrices are ill-conditioned, because the convergence rate depends on the condition number of the Jacobian operator for the factorization and the Hessian of the loss function with respect to the weight matrix. To address this challenge, we adopt the Riemannian gradient descent (RGD) algorithm on the Riemannian manifold of fixed-rank matrices to update the entire low-rank weight matrix. This algorithm completely avoids the factorization, thereby eliminating the negative impact of the Jacobian condition number.

There is extensive literature on accelerating first-order optimization methods in an Euclidean setting. Under which conditions such acceleration is feasible in Riemannian optimization problems is an active area of research. Motivated by the recent success of silver stepsize methods in the Euclidean setting, we undertake a study of such algorithms in the Riemannian setting. We provide the new class of algorithms determined by the choice of vector transport that allows the silver stepsize acceleration on Riemannian manifolds for the function classes associated with the corresponding vector transport. As a core application, we show that our algorithm recovers the standard Wasserstein gradient descent on the 2-Wasserstein space and, as a result, provides the first provable accelerated gradient method for potential functional optimization problems in the Wasserstein space.