ISL Colloquium
The ISL Colloquium convenes in person again (on Thursdays at 4pm PT in Packard 101) as of Fall 2021. For those unable to attend, talks are also streamed via Zoom. To avoid "Zoom-bombing", we ask attendees to input their email address here https://stanford.zoom.us/meeting/register/tJckfuCurzkvEtKKOBvDCrPv3McapgP6HygJ to receive the Zoom meeting details via email.

← List all talks ...

Computational Barriers to Estimation from Low-Degree Polynomials

Alex Wein – Instructor, NYU's Courant Institute

Thu, 8-Oct-2020 / 4:30pm / Zoom: https://stanford.zoom.us/meeting/register/tJckfuCurzkvEtKKOBvDCrPv3McapgP6HygJ

Talk

Abstract

One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for such problems. Many leading algorithmic paradigms (such as spectral methods and approximate message passing) can be captured by low-degree polynomials, and thus, lower bounds against low-degree polynomials serve as evidence for computational hardness of statistical problems.

Prior work has studied the power of low-degree polynomials for the detection (i.e. hypothesis testing) task. In this work, we extend these methods to address problems of estimating (i.e. recovering) the planted signal instead of merely detecting its presence. For a large class of “signal plus noise” problems, we give a user-friendly lower bound for the best possible mean squared error achievable by any degree-D polynomial. These are the first results to establish low-degree hardness of recovery problems for which the associated detection problem is easy. As applications, we study the planted submatrix and planted dense subgraph problems, resolving (in the low-degree framework) open problems about the computational complexity of recovery in both cases.

Joint work with Tselil Schramm, available at: https://arxiv.org/abs/2008.02269

Bio

Alex Wein is a Courant Instructor (postdoc) in the department of mathematics at NYU’s Courant Institute. He received a PhD from the MIT department of mathematics in 2018, advised by Ankur Moitra. His research interests are centered around understanding the computational complexity of problems in high-dimensional statistics.