Duc Toan Nguyen

Nonnegative Matrix Factorization (NMF) is a dimensionality reduction technique used in data analysis and machine learning. It aims to decompose a nonnegative matrix as a product of two lower-rank nonnegative matrices. In this project, we implemented and compared multiple minimum-volume NMF algorithms in synthetic and practical data. Also, we applied the Majorization-Minimization idea, a popular paradigm for solving challenging optimization problems. We proposed a new algorithm named Square-Root Minimum-Volume NMF inspired by the Square-Root Lasso problem. Our method improves the previous minimum-volume NMF methods by automatically determining the optimal tuning parameter. I presented this research at the Gulf Coast Undergraduate Research Symposium (GCURS) 2023 and received the Outstanding Session Presentation Award from the Computational Applied Mathematics and Operations Research Department, Rice University. Our paper is published in the Proceedings of the 2024 SIAM International Conference on Data Mining (SDM24). Moreover, I also achieved the SIAM Student Travel Award for the SDM24!

Please visit these links for:
 - The paper.
 - The presentation slides
 - The poster

I worked on this project with Dr. Eric C. Chi under REU STAT-DATASCI 2023 at Rice Department of Statistics.

News!!!:
  - I will present this project at Thursday Poster Session (PP4), SIAM Conference on Mathematics of Data Science (MDS24) (abstract)
  - I will also present this project at AMS Contributed Papers Session, Numerical analysis I, Joint Mathematics Meetings (JMM 2025) (abstract)

Variance Reduction is an approach in Stochastic Optimization Methods that improves the convergence rate compared to the original Stochastic Gradient Descent (SGD). Intuitively, Variance Reduction methods decrease the variance of the gradient so that the gradient is more direct to the optimal point. Moreover, the shuffling paradigm has recently become a 'hot' topic in the optimization society, and it can improve the rate for SGD in many cases. In this research project, we combine the shuffling paradigm into a famous variance reduction method called SARAH (StochAstic Recursive grAdient algoritHm), analyze its efficiency (convergence rate and complexity), and compare it to other shuffling gradient methods.

I am working on this project with Dr. Lam M. Nguyen (IBM Research) and Dr. Trang H. Tran (Lehigh ISE).

A geodesic net is a network that connects multiple points with the geodesics while ensuring that each additional point is " balanced " and stretched equally by its neighbors through those curves. In this project, we are finding a new algorithm that can construct geodesic nets on different surfaces based on Steiner trees since Steiner trees are satisfied to be geodesic nets. Our algorithm is inspired by the construction of Steiner Tree by bubble soap film, illustrated in the videos below. Moreover, we are finding a condition for the existence of a balanced point in a triangle on a general surface. The investigation is divided into two cases: surfaces with non-positive curvature and surfaces with positive curvature.
 - In the non-positive curvature case, our proof is based on the Gauss-Bonnet theorem.
 - In the positive curvature case, we find a condition for the existence of a balanced point on a general sphere with curvature \(1/R^2\).
Then, we use Jacobi fields (vector fields describing how geodesics "spread") and Rauch comparison theorem to find a general condition on a surface with curvature bounded above by \(1/R^2\) based on the condition on the sphere.

I am working on this project with Dr. Ken Richardson at TCU Department of Mathematics.
Please try our algorithm at https://github.com/ductoanng/BubbleGeodesicNet.

News!!!:
  - Here is the preprint version for the paper (link)
  - I will present this project at AMS - PME Undergraduate Student Poster Session, Joint Mathematics Meetings (JMM 2025) (abstract) (poster)
  - I also received the Pi Mu Epsilon (PME) Student Travel Funding to present this project

GO2AI is a project working with AI and the game Go to optimize playing. It is inspired by the advent of AlphaGo 2016. In this project, we aim to train AI "slowly" to discover the similarities and differences between AI and human learning processes. We are applying Grad-CAM, an explainable AI method, to understand better how the neural network influences the policy of the AI agent in the decision-making process. The image below shows how the grad-CAM method visualizes how model Q (the AI agent) decides the optimal move from iteration 1 to the final iteration (iter 70). I am part of a dedicated team, including Blake Good, Harrison Leath, Shawn Fahimi, and me, under the supervision of Dr. Liran Ma and Dr. Ze-li Dou at the TCU Department of Computer Science.