arXiv subjects

Duc Toan Nguyen

Publications and source records attributed to Duc Toan Nguyen.

5 recordsLinked to original sources

arXiv2026

Adjusted Shuffling SARAH: Advancing Complexity Analysis via Dynamic Gradient Weighting

In this paper, we propose Adjusted Shuffling SARAH, a novel algorithm that integrates shuffling strategies into the recursive SARAH framework using a dynamic weighting mechanism to enhance exploration. We analyze the algorithm under two operating modes. First, we show that the Exact Mode matches the best-known theoretical guarantees for shuffling variance-reduced methods in both strongly convex and non-convex settings. Second, to address large-scale regimes, we introduce an Inexact Mode that utilizes mini-batch estimators. A key contribution of our work is proving that this Inexact Mode achieves a total complexity independent of the dataset size, making it significantly more scalable than existing shuffling methods when the sample size is large.

math.OC↗

arXiv2026

Fréchet Regression on the Bures-Wasserstein Manifold

Fréchet regression, or conditional Barycenters, is a flexible framework for modeling relationships between covariates (usually Euclidean) and response variables on general metric spaces, e.g., probability distributions or positive definite matrices. However, in contrast to classical barycenter problems, computing conditional counterparts in many non-Euclidean spaces remains an open challenge, as they yield non-convex optimization problems with an affine structure. In this work, we study the existence and computation of conditional barycenters, specifically in the space of positive-definite matrices with the Bures-Wasserstein metric. We provide a sufficient condition for the existence of a minimizer of the conditional barycenter problem that characterizes the regression range of extrapolation. Moreover, we further characterize the optimization landscape, proving that under this condition, the objective is free of local maxima. Additionally, we develop a projection-free and provably correct algorithm for the approximate computation of first-order stationary points. Finally, we provide a stochastic reformulation that enables the use of off-the-shelf stochastic Riemannian optimization methods for large-scale setups. Numerical experiments validate the performance of the proposed methods on regression problems of real-world biological networks and on large-scale synthetic Diffusion Tensor Imaging problems.

math.OC↗

arXiv2026

Intrinsic Decentralized Stochastic Riemannian Optimization on Manifolds with Bounded Sectional Curvature

Decentralized optimization on Riemannian manifolds is foundational for many modern machine learning and signal processing applications in which data are non-Euclidean and generated and processed in a distributed manner. Although intrinsic Riemannian methods exploit manifold geometry without relying on Euclidean embeddings, existing decentralized Riemannian optimization algorithms typically use constant step sizes and therefore converge only to a neighborhood of steady-state error. In this paper, we study the decentralized stochastic Riemannian gradient method in the diminishing step-size regime on manifolds with (possibly positive) bounded sectional curvature. We prove an $O(1/T)$ bound for the network consensus error and an $O(\log T/\sqrt{T})$ ergodic bound for the global optimality gap. To the best of our knowledge, this is the first exact, non-asymptotic optimality-gap guarantee for an intrinsic decentralized stochastic Riemannian method in the geodesically convex setting. Furthermore, the diminishing step-size schedule allows substantially larger initial gradient steps than fixed-step baselines, leading to better performance in practice. We illustrate this on the problem of distributed PCA over a Grassmann manifold.

math.OC↗

arXiv2025

On the existence of a balanced vertex in geodesic nets with three boundary vertices

Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than $2π/3$, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle.

math.DG↗

arXiv2023

Towards Tuning-Free Minimum-Volume Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) is a versatile and powerful tool for discovering latent structures in data matrices, with many variations proposed in the literature. Recently, Leplat et al.\@ (2019) introduced a minimum-volume NMF for the identifiable recovery of rank-deficient matrices in the presence of noise. The performance of their formulation, however, requires the selection of a tuning parameter whose optimal value depends on the unknown noise level. In this work, we propose an alternative formulation of minimum-volume NMF inspired by the square-root lasso and its tuning-free properties. Our formulation also requires the selection of a tuning parameter, but its optimal value does not depend on the noise level. To fit our NMF model, we propose a majorization-minimization (MM) algorithm that comes with global convergence guarantees. We show empirically that the optimal choice of our tuning parameter is insensitive to the noise level in the data.

stat.ML↗