题目：Can one Hear the Shape of High-Dimensional Landscape?

报告人：王式柔 教授 （吉林大学）

地点：致远楼101室

时间：2025年10月27日10:00-11:00

摘要：Potential functions used in optimizations, dynamics applications, and machine learning etc. can be rather complicated in term of their structures and properties especially in very high dimensions. Due to lacking of knowledge on concrete forms of potential functions in real applications, even the determination of their basic structures and properties is a challenging problem in both mathematical analysis and numerical simulations. This talk presents a probabilistic approach to investigate the landscape of potential functions, including those in high dimensions, by using an appropriate coupling scheme to couple two copies of the overdamped Langevin dynamics of the potential functions. It can be theoretically shown that for potential functions with single or multiple wells, the coupling time distributions admit qualitatively distinct exponential tails in terms of noise magnitudes. Moreover, a novel quantity is introduced which characterizes the non-convexity of a multi-well potential function. These theoretical findings suggest a promising approach to probe the shape of a potential landscape through the coupling time distributions. As an illustrative example, numerical results of the loss landscapes of neural networks of different sizes will be presented. This talk is based on joint work with Yao Li (UMass) and Molei Tao (Georgia Tech).

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