The literature for the geometric properties of Riemannian Manifolds is immense and hence we cannot hope to survey them here; for an appetizer, we refer the reader to Burago et al. [93] and Lee [94] and references therein. On the other hand, as stated, it is not until recently that the long-run non-asymptotic behavior of optimization algorithms in Riemannian manifolds (even the smooth ones) has encountered a lot of interest. For concision, we have deferred here a detailed exposition of the rest of recent results to Appendix A of the paper's supplement. Additionally, in Appendix B we also give a bunch of motivating examples which can be solved by Riemannian min-max optimization. Many application problems can be formulated as the minimization or maximization of a smooth function over Riemannian manifold and has triggered a line of research on the extension of the classical first-order and second-order methods to Riemannian setting with asymptotic convergence to first-order stationary points in general [95].

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.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and interpretable concept that captures intrinsic geometric structure and underpins numerous tasks, from community detection to geometric deep learning. A wide range of discrete curvature models have been proposed for various data representations, including graphs, simplicial complexes, cubical complexes, and point clouds sampled from manifolds. These models not only provide efficient characterizations of data geometry but also constitute essential components in geometric learning frameworks. In this paper, we present the first comprehensive review of existing discrete curvature models, covering their mathematical foundations, computational formulations, and practical applications in data analysis and learning. In particular, we discuss discrete curvature from both Riemannian and metric geometry perspectives and propose a systematic pipeline for curvature-driven data analysis. We further examine the corresponding computational algorithms across different data representations, offering detailed comparisons and insights. Finally, we review state-of-the-art applications of curvature in both supervised and unsupervised learning. This survey provides a conceptual and practical roadmap for researchers to gain a better understanding of discrete curvature as a fundamental tool for geometric understanding and learning.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. One of the most common reasons for using Markov chain Monte Carlo (MCMC) is to estimate the value of an otherwise intractable integral. Typically MCMC algorithms will give an exact answer as the number of iterations increases to infinity. However, this gives little assurance about the precision of the estimate in finite samples. This paper addresses this important issue from a theoretical point of view for the Hamiltonian Monte Carlo (HMC) algorithm, an algorithm which has been receiving a fair amount of recent attention.