Academic Report of School of Mathematical Sciences [2026] No. 039

(Series Report for High-Level University Construction No. 1298)

Title：Riemannian Momentum on Quaternionic Manifolds: A New Frontier for High-Dimensional Signal Recovery

Speaker：Mingqing Xiao, Professor (Southern Illinois University)

Time：14:30-16:30, May. 13, 2026

Location：Huixing Building Room 702

Abstract：Quaternionic signal processing has emerged as a vital framework for handling multi-dimensional data, particularly in color image recovery and computer vision, where the non-commutative structure of quaternions naturally preserves cross-channel correlations. However, as we move toward large-scale data, we face a fundamental conflict: how do we leverage the acceleration of momentum-based methods like the Quaternionic Randomized Kaczmarz (QRKAM) while strictly adhering to the low-rank structures that define physical systems?

In traditional Euclidean frameworks, momentum updates generically "inflate" the rank of iterates, leading to a computational bottleneck where a truncated QSVD is required at every step. This overhead renders many state-of-the-art methods impractical for the very high-dimensional problems they were designed to solve.

In this talk, we present a transformative approach: the Riemannian Quaternion Regularized Kaczmarz method with Adaptive Momentum. By shifting the optimization from the ambient Euclidean space to the intrinsic geometry of the fixed-rank quaternion manifold, we eliminate rank inflation by construction. We will discuss:

• The Geometry of Skew-Fields: The derivation of Riemannian components—tangent spaces, metrics, and retractions—specifically tailored for the non-commutative quaternion skew-field.

• Adaptive Curvature Correction: A new class of adaptive step-size and momentum parameters that incorporate a "curvature correction" term, allowing the optimizer to navigate the manifold's sectional geometry.

• Convergence and Scalability: Theoretical proof of expected linear convergence, demonstrating that our Riemannian approach reduces computational costs by orders of magnitude in high-dimensional settings.

We conclude by demonstrating how this manifold-aware framework provides a robust and scalable solution for the next generation of quaternionic imaging and signal recovery tasks.

Speaker Profile：Dr. Mingqing Xiao studied at the University of Illinois at Urbana-Champaign (UIUC) from 1991 to 1997. He worked under the guidance of Tamer Basar, a renowned expert in control theory and a member of the National Academy of Engineering, conducting research in robust control theory. He received his Ph.D. in Applied Mathematics in 1997. From July 1997 to December 1999, he served as an assistant professor at the University of California, Davis. Since 2000, he has been on the faculty at Southern Illinois University, where he served as an assistant professor (2000–2002), associate professor (2002–2007), and full professor (2007–present). He served as a visiting researcher at the U.S. Air Force Research Laboratory (Wright-Patterson Air Force Base) from 2001 to 2002. He is currently a doctoral advisor in the Graduate School at Southern Illinois University, Director of Graduate Studies in the Department of Mathematics and Statistics, a member of the University Senate, and a reviewer for the university’s research grants.

Dr. Mingqing Xiao has published over 150 journal articles, conference papers, and monographs in the fields of big data analysis, robot learning, optimization theory, numerical computation of partial differential equations, and control theory. Dr. Xiao’s research has been supported by the National Science Foundation (NSF), the Qatar National Research Fund, and the U.S. Air Force Office of Scientific Research. In 2003, he co-edited the monograph New Trends in Nonlinear Dynamics and Control and their Applications (Springer-Verlag, 2003), which was funded by the National Science Foundation and the U.S. Air Force Office of Scientific Research. Since 2010, Professor Xiao has served as a member of the organizing committee for the SIAM Conference on Control and Its Applications. He also served on the organizing committee for the 3rd World Congress on Optimization (2013). He also served as an academic editor for the 18th IFAC (International Federation of Automatic Control) World Congress (2011, Milan, Italy) and as a guest editor for Computational Intelligence and Information Security, Computational Innovation and Control Information (2010), Mathematical Problems in Engineering (2011, 2012), and the Journal of the Franklin Institute (2014). He has served on the editorial boards of many renowned academic journals, including IEEE Transactions on Automation (IEEE TAC, 2004–2007), Automatica (2008–2011), European Journal of Control (2010–present), Numerical Algebra, Control, and Optimization (2010–2018), and Nonlinear Analysis: Hybrid Systems (2011–2015), among other specialized journals. He has also served as a reviewer for the U.S. National Science Foundation (2019–2021), the U.S.-Israel Binational Science Foundation (2010–present), and the Romanian National Council for Scientific Research (2011–present). He has also been a keynote speaker at numerous academic conferences. In 2016, he was named Distinguished Scholar by the College of Science at Southern Illinois University. In 2026, he received the Juh Wah Chen Distinguished Award from the College of Engineering at Southern Illinois University.

Faculty and students are welcome to attend!

Invited by: Xiaoli Sun

School of Mathematical Sciences

May 7, 2026