arXiv research

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

Optimal algorithms for Riemannian optimization with reduced complexity.

Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.

The paper analyzes convergence of Riemannian SA schemes for stochastic optimization.

The paper analyzes fixed step-size SA schemes on Riemannian manifolds.

New method finds near-optimal solutions for non-convex optimization problems.

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.

Paper shows $L^\infty$-positivity and stochastic completeness are equivalent.

A simpler metric for latent space geometry.

New method for sampling diffusion bridges on sub-Riemannian manifolds.

In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we he…

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…

New geometric SDEs and discretizations on Riemannian manifolds with error bounds.

We propose a stochastic gradient Markov chain Monte Carlo (SG-MCMC) algorithm for scalable inference in mixed-membership stochastic blockmodels (MMSB). Our algorithm is based on the stochastic gradient Riemannian Langevin sampler and achieves both faster speed and higher accuracy at every iteration than the current sta…

New method estimates Riemannian derivatives from noisy function evaluations.

SMAVE optimizes SDR by projecting onto a low-dimensional subspace on a Riemannian manifold.

We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …

Method approximates Riemannian barycenter on manifolds.

Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.

The paper proposes a novel method for optimizing bounded functions using Fourier series and Ricci flow.

Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.

Study approximates Riemannian manifolds using polyhedra.

The paper calculates how random changes affect paths on a complex geometric space.

The article constructs stochastic integration in Riemannian manifolds.

The book explores stochastic areas and heat kernels on manifolds.

The SABR model is a stochastic volatility model not admitting a closed form solution. Hagan, Kumar, Leniewski and Woodward have obtained an approximate solution by means of perturbative techniques. A more precise approximation was found by Henry-Labordère with the heat kernel expansion method. The latter relies on deep…

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

This paper extends Mirror Descent to Riemannian manifolds for optimization.

High-probability bound for distributed stochastic approximation tracking error.

Functional-analytic method for stochastic parallel transport in bundles.

Two new methods solve nonsmooth optimization on Riemannian Stiefel manifold.

Stochastic approximation algorithms show exponential progress bounds.

We study the stochastic Riemannian gradient algorithm for matrix eigen-decomposition. The state-of-the-art stochastic Riemannian algorithm requires the learning rate to decay to zero and thus suffers from slow convergence and sub-optimal solutions. In this paper, we address this issue by deploying the variance reductio…

Deviation inequalities for stochastic approximation methods.

Neural networks can approximate complex stochastic equations well.

New algorithms for approximating stochastic processes efficiently.

Paper develops a finite dimensional approximation scheme for Riemannian manifolds.

The paper develops stochastic methods on geometric spaces for transformations.

We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of …

This work analyzes nonexpansive stochastic approximations with Markovian noise, proving convergence in reinforcement learning.

Paper develops SINNOs for approximating stochastic processes.

RNGI model bridges two probability densities on Riemannian manifolds efficiently.

Stochastic gradient descent on manifolds improves low-rank approximation.

Universal approximation for stochastic processes using Brownian motion.

We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood math…

Study on local convergence of min-max algorithms to differential equilibria on Riemannian manifolds.

Approximates derivative pricing under fractional stochastic volatility.