Speaker: 鲍志刚副教授

Title: Phase transition for the bottom singular vector of rectangular random matrices

Language: Chinese

Time & Venue: 2024.12.27 10:00-11:00 数学院N620

Abstract: We consider the rectangular random matrix X=(x_{ij})\in \mathbb{R}^{N\times n} whose entries are iid with tail probability t^{-\alpha} for some \alpha>0. We consider the regime N(n)/n\to a>1 as n tends to infinity. Our main focus is the right singular vector corresponding to the smallest singular value, which will be called the "bottom singular vector". In this talk, we show the following phase transition regarding the localization length of the bottom singular vector: when \alpha<2 the localization length is O(n/\log n); when \alpha>2 the localization length is of order n. The variational definition of the bottom singular vector suggests that the mechanism for this localization-delocalization transition when \alpha goes across 2 is intrinsically different from the one for the top singular vector when \alpha goes across 4. This is a joint work with Jaehun Lee and Xiaocong Xu.