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<div class="" style="word-wrap:break-word"><span class="" style="font-size:14.666666984558105px">This is an announcement of Goutham Rajendran's Dissertation Defense.</span><br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">===============================================</span><br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">Candidate: Goutham Rajendran</span><br class="" style="font-size:14.666666984558105px">
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<span class="" style="font-size:14.666666984558105px">Date: Monday, June 13, 2022</span><br class="" style="font-size:14.666666984558105px">
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<span class="" style="font-size:14.666666984558105px">Time: 11 am CST</span><br class="" style="font-size:14.666666984558105px">
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<span class="" style="font-size:14.666666984558105px">Remote Location: </span><a href="https://uchicago.zoom.us/j/96784507440?pwd=VlhqVHVJb3p2NStQZWF5aFBXeW1lUT09" class="" style="font-size:14.666666984558105px">https://uchicago.zoom.us/j/96784507440?pwd=VlhqVHVJb3p2NStQZWF5aFBXeW1lUT09</a><span class="" style="font-size:14.666666984558105px"> Meeting
ID: 967 8450 7440 Passcode: 670915</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">Location: JCL 346</span><br class="" style="font-size:14.666666984558105px">
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<span class="" style="font-size:14.666666984558105px">Title: Nonlinear Random Matrices and Applications to the Sum of Squares Hierarchy</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">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.</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">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.</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">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.</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">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.</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">Advisors: Madhur Tulsiani and Aaron Potechin</span><br class="" style="font-size:14.666666984558105px">
<br class="" style="font-size:14.666666984558105px">
<span class="" style="font-size:14.666666984558105px">Committee Members: Madhur Tulsiani, Aaron Potechin, and Janos Simon</span><br class="" style="font-size:14.666666984558105px">
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