June 2022 Computational barriers to estimation from low-degree polynomials
Tselil Schramm, Alexander S. Wein
Author Affiliations +
Ann. Statist. 50(3): 1833-1858 (June 2022). DOI: 10.1214/22-AOS2179


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: it has been demonstrated in various settings that low-degree polynomials of the data can match the statistical performance of the best known polynomial-time algorithms. Prior work has studied the power of low-degree polynomials for the task of detecting the presence of hidden structures. In this work, we extend these methods to address problems of estimation and recovery (instead of detection). 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. To our knowledge, these are the first results to establish low-degree hardness of recovery problems for which the associated detection problem is easy. As applications, we give a tight characterization of the low-degree minimum mean squared error for 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.

Funding Statement

ASW was partially supported by NSF Grant DMS-1712730 and by the Simons Collaboration on Algorithms and Geometry.


For helpful discussions, we are grateful to Afonso Bandeira, Matthew Brennan, Jingqiu Ding, David Gamarnik, Sam Hopkins, Frederic Koehler, Tim Kunisky, Jerry Li, Jonathan Niles-Weed and Ilias Zadik. We thank the anonymous reviewers for their helpful comments.

This work was done while TS was virtually visiting the Microsoft Research Machine Learning and Optimization group.


Download Citation

Tselil Schramm. Alexander S. Wein. "Computational barriers to estimation from low-degree polynomials." Ann. Statist. 50 (3) 1833 - 1858, June 2022. https://doi.org/10.1214/22-AOS2179


Received: 1 August 2020; Revised: 1 September 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441142
zbMATH: 07547952
Digital Object Identifier: 10.1214/22-AOS2179

Primary: 62F15
Secondary: 68Q17

Keywords: computational complexity , Low-degree polynomials , planted dense subgraph , planted submatrix

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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