Counterexamples to the Low-Degree Conjecture

Authors Justin Holmgren, Alexander S. Wein

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Justin Holmgren
  • NTT Research, Palo Alto, CA, USA
Alexander S. Wein
  • Courant Institute of Mathematical Sciences, New York University, NY, USA


We thank Sam Hopkins and Tim Kunisky for comments on an earlier draft.

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Justin Holmgren and Alexander S. Wein. Counterexamples to the Low-Degree Conjecture. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 75:1-75:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Low-degree likelihood ratio
  • error-correcting codes


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