ISE Seminar Series - Zhang

Abstract: Low-rank structures permeate a vast array of real-world applications, from machine learning models to power grid optimization to scheduling and planning in industrial engineering. Their power lies in their natural ability to encode high-dimensional phenomena through low-dimensional latent representations and to represent discrete integer decisions as continuous surrogates.

However, the inherent nonconvexity of low-rank optimization has long been seen as a prohibitive barrier: direct nonconvex methods risk getting trapped in spurious local minima, while convex relaxations often result in intractable computations at scale.

In this talk, we challenge this conventional view by embracing nonconvexity and mitigating its drawbacks. We show that overparameterizing the low-rank factorization systematically reduces the occurrence of spurious local minima through a stepwise process, where higher ranks progressively "smooth out" the landscape until every local minimum becomes global and every saddle point is escapable.

We also present a posteriori certification methods based on rank deficiency to verify global optimality post-optimization, and introduce an inexpensive preconditioner that restores the linear convergence of gradient descent in overparameterized regimes.

Finally, we discuss how these advances lay the groundwork for developing low-rank optimization as a distinct and impactful discipline within mathematical optimization—one that bridges theory, computation, and real-world applications in safety-critical domains.

Bio: Richard Y. Zhang is an Assistant Professor of Electrical and Computer Engineering at the University of Illinois Urbana-Champaign. He received the B.E. (hons) degree with first class honours from the University of Canterbury, New Zealand, and earned his Ph.D.

in Electrical Engineering and Computer Science at MIT. Before joining Illinois, he was a Postdoctoral Scholar at UC Berkeley. His research focuses on low-rank optimization, both as a theoretical framework for understanding how algorithms uncover latent structure in complex data, and as a computational approach for scaling algorithms to massive problems. He received the NSF CAREER Award in 2021, and serves as Area Chair at NeurIPS, ICML, and ICLR.