The Southern California Probability Symposium will take place, Saturday, April 5 here at UCI. It will start with a continental breakfast at 9:00 am in ISEB 1300 and run until 5:30pm. Here's a link to the symposium web page: https://scps.pstat.ucsb.edu/SCPS2025.html
Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, the study of elections and social choice, and graph theory. We will give an introduction to the topic (with an emphasis on robust statistics), describe some remaining open questions as well as our recent progress towards the solutions.
This talk is based on the joint work with Yinan Shen.
Abstract: In this talk, we present our results for the 1-bit phase retrieval problem: the recovery of a possibly sparse signal $x$ from the observations $y=sign(|Ax|-1)$. This problem is closely related to 1-bit compressed sensing where we observe $y=sign(Ax)$, and phase retrieval where we observe $y=|Ax|$. We show that the major findings in 1-bit compressed sensing theory, including hyperplane tessellation, optimal rate and a very recent efficient and optimal algorithm, can be developed in this phase retrieval setting. This is a joint work with Ming Yuan: https://arxiv.org/abs/2405.04733.
In this talk, we present a new proof of the delocalization conjecture for random band matrices. The conjecture asserts that for an $N\times N$ random matrix with bandwidth $W$, the eigenvectors in the bulk of the spectrum are delocalized provided that $W \gg N^{1/2}$. Moreover, in this regime, the eigenvalue distribution aligns with that of Gaussian random ensembles (i.e., GOE or GUE). Our proof employs a novel loop hierarchy method and leverages the sum-zero property, a concept that was instrumental in the previous work on high-dimensional random matrices.
