Duc Toan Nguyen

Research question: Fréchet regression models Euclidean predictors and metric-space responses, but on the Bures-Wasserstein manifold its signed weights can make extrapolation ill-posed or produce invalid positive-definite predictions. What conditions make these conditional Bures-Wasserstein barycenters exist and remain computable?

Main contributions: The paper gives a spectral-dominance condition guaranteeing a minimizer and characterizes the objective's landscape. It also develops projection-free Riemannian and pairwise stochastic methods, with experiments on network regression and large-scale diffusion tensors.

Paper: accepted at NeurIPS 2026.

Advisor: Dr. César A. Uribe (Department of Electrical and Computer Engineering, Rice University).

Research question: Decentralized learning on non-Euclidean data needs methods that respect manifold geometry; existing intrinsic methods often use fixed step sizes and converge only to a neighborhood. Can diminishing step sizes give stronger guarantees on manifolds with bounded sectional curvature, including positive curvature?

Main contributions: The paper proves an \(O\!\left(\frac{1}{T}\right)\) network-consensus error bound and an \(O\!\left(\frac{\log T}{\sqrt{T}}\right)\) ergodic optimality-gap bound, giving an exact non-asymptotic guarantee for this setting. It demonstrates the method on distributed PCA over a Grassmann manifold.

Paper: IEEE Control Systems Letters, 10 (2026), and invited session presentation at IEEE CDC 2026.

Advisor: Dr. César A. Uribe (Department of Electrical and Computer Engineering, Rice University).

Research question: Nonnegative matrix factorization represents nonnegative data with lower-rank nonnegative factors, while minimum-volume variants rely on a tuning parameter. Can that parameter be selected automatically without giving up the minimum-volume approach?

Main contributions: The paper proposes Square-Root Minimum-Volume NMF, inspired by the Square-Root Lasso, and uses a majorization-minimization algorithm to remove the need to tune the parameter manually. The method is evaluated on synthetic and practical data.

Paper: Proceedings of the 2024 SIAM International Conference on Data Mining (SDM 2024).

Advisor: Dr. Eric C. Chi (School of Statistics, University of Minnesota).

Research question: Shuffling can improve stochastic-gradient methods, but variance-reduced methods must balance strong convergence guarantees with the cost of full-gradient computations. How can shuffling be integrated into SARAH while scaling to large datasets?

Main contributions: The paper introduces a dynamic-weighting shuffling variant of SARAH and analyzes exact and mini-batch inexact modes. The exact mode matches best-known guarantees in strongly convex and non-convex settings, while the inexact mode has total complexity independent of dataset size.

Paper: Journal of Optimization Theory and Applications (JOTA), 2026.

Advisors: Dr. Trang H. Tran (Emory University) and Dr. Lam M. Nguyen (IBM Research).

Research question: A geodesic net has geodesic edges and balanced interior vertices, where the incident unit tangent vectors sum to zero. Under what geometric conditions does a triangle on a general Riemannian surface contain such a balanced vertex?

Main contributions: The paper proves existence when the triangle's angles are each below \(2\pi/3\) and its side lengths are sufficiently small, generalizing the Fermat-point result from the plane. The broader project also investigates construction of geodesic nets on surfaces.

Paper: Journal of Geometry, 116 (2025), article 36.

Advisor: Dr. Ken Richardson (Department of Mathematics, Texas Christian University).

Research question: Can training a Go-playing AI gradually reveal similarities and differences between human and machine learning?

Main contributions: The project applies Grad-CAM to visualize how the agent's neural network influences move selection over training, including its progression from early iterations to iteration 70. It was conducted with Blake Good, Harrison Leath, and Shawn Fahimi under Dr. Liran Ma and Dr. Ze-li Dou at TCU.

Project: GO2AI at TCU.

TCU team: Blake Good, Harrison Leath, and Shawn Fahimi; advisors Dr. Liran Ma and Dr. Ze-li Dou.