Gaussian comparison inequalities are classical tools that often lead to simple proofs of powerful results in random matrix theory, convex geometry, etc. Perhaps the most celebrated of these tools is Slepian’s Inequality, which dates back to 1962. The Gaussian Min-max Theorem (GMT) is a non-trivial generalization of Slepian’s result, derived by Gordon in 1988. Here, we prove a tight version of the GMT in the presence of convexity. Based on that, we describe a novel and general framework to precisely evaluate the performance of non-smooth convex optimization methods under certain measurement ensembles (Gaussian, Haar). We discuss applications of the theory to box-relaxation decoders in massive MIMO, 1-bit compressed sensing, and phase-retrieval.
In this talk, we first discuss existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representation. We then talk about sharp two- sided estimates for fundamental solutions of general time fractional equations in metric measure spaces. This is a joint work with Zhen-Qing Chen(University of Washington, USA), Takashi Kumagai (RIMS, Kyoto University, Japan) and Jian Wang (Fujian Normal University, China).
The invertibility of random matrices with iid entries has been the object of intense study over the past decade, due in part to its role in proving the circular law, as well as its importance in numerical analysis (smoothed analysis). In this talk we review recent progress in our understanding of invertibility for some non-iid models: adjacency matrices of sparse random regular digraphs, and random matrices with inhomogeneous variance profile. We will also discuss estimates for the number of singular values in short intervals. Graph regularity properties play a key role in both problems. Based in part on joint works with Walid Hachem, Jamal Najim, David Renfrew, Anirban Basak and Ofer Zeitouni.
Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and then present new results on the problem of data classification from binary data that proposes a stochastic framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, provide a theoretical analysis for a simple case, and discuss future directions.