[Colloquium] Goutham Rajendran Dissertation Defense/Jun 13, 2022

[Megan Woodward]

This is an announcement of Goutham Rajendran's Dissertation Defense.
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Candidate: Goutham Rajendran

Date: Monday, June 13, 2022

Time: 11 am CST

Remote Location: https://uchicago.zoom.us/j/96784507440?pwd=VlhqVHVJb3p2NStQZWF5aFBXeW1lUT09 Meeting ID: 967 8450 7440 Passcode: 670915

Location: JCL 346

Title: Nonlinear Random Matrices and Applications to the Sum of Squares Hierarchy

Abstract: We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum-of-Squares (SoS) hierarchy on average-case problems.

The SoS hierarchy is a powerful optimization technique that has achieved tremendous success for various problems in combinatorial optimization, robust statistics and machine learning. It's a family of convex relaxations that lets us smoothly tradeoff running time for approximation guarantees. In recent works, it's been shown to be extremely useful to recover structure in high dimensional noisy data. It also remains our best approach towards refuting the notorious Unique Games Conjecture.

In this work, we analyze the performance of the SoS hierarchy on fundamental problems stemming from statistics, theoretical computer science and statistical physics. In particular, we show subexponential-time SoS lower bounds for the problems of the Sherrington-Kirkpatrick Hamiltonian, Planted Slightly Denser Subgraph, Tensor Principal Components Analysis and Sparse Principal Components Analysis. These SoS lower bounds involve analyzing large random matrices, wherein lies our main contributions. These results offer strong evidence for the truth of and insight into the low-degree likelihood ratio hypothesis, an important conjecture that predicts the power of bounded time algorithms for hypothesis testing.

We also develop general-purpose tools for analyzing the behavior of random matrices which are functions of independent random variables. Towards this, we build on and generalize the matrix variant of the Efron-Stein inequalities. In particular, our general theorem on matrix concentration recovers various results that have appeared in the literature. We expect these random matrix theory ideas to have other significant applications.

Advisors: Madhur Tulsiani and Aaron Potechin

Committee Members: Madhur Tulsiani, Aaron Potechin, and Janos Simon




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