The Geometry of Signal Recovery

Part of a series of seminars at the Dept. of Mathematical Sciences, UNLV.

Speaker: Justin Le.

Host: Prof. David Costa.

Friday, Nov. 17, 2017

11:30am to 1:30pm

CBC C-222, UNLV

Abstract

In this talk, we discuss a convex program for recovering a structured signal from random linear measurements. Our emphasis will be on the derivation of bounds on estimation error that are afforded by Gaussian randomness of the measurements. In particular, the use of Gordon's "escape" lemma allows us to compute bounds on the minimum conic singular value associated with the measurement operator and with the descent cone of a complexity measure that encourages a certain structure on the recovered signal. We discuss the role of Gaussian width (and statistical dimension), as well as recent efforts toward non-Gaussian and non-linear cases of the problem.

Program

Overview

• Cones, complexity, and general bounds
• Bounds via the Gordon-Slepian lemma and Gaussian concentration
• Notions of restrictedness of operators
• Algorithmic considerations; remarks

Suggested Reading

• R. Vershynin. Estimation in high dimensions: a geometric perspective. Arxiv. 2014.
• K. R. Davidson and S. J. Szarek. Local operator theory, random matrices, and Banach spaces. Handbook of the Geometry of Banach Spaces, Volume 1. 2001.
• V. Chandrasekaran, et al. The convex geometry of linear inverse problems. Foundations of Computational Mathematics. 2012.
• S. Oymak and J. Tropp. Universality laws for randomized dimension reduction. Arxiv. 2017.
• Y. Gordon. On Milman's inequality and random subspaces which escape through a mesh in Rⁿ. Geometric Aspects of Functional Analysis.. 1988.
• M. Ledoux and M. Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Springer, 1991 (reprint 2011).

No slides are available. The talk is given on whiteboard to facilitate interaction with the audience when discussing the proofs.