Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent
Matthew Brennana, Guy Breslerb, Samuel B. Hopkinsc, Jerry Zheng Lid, Tselil Schramme
- aHarvard University, Cambridge, United States
- bMassachusetts Institute of Technology, Cambridge, United States
- cUniversity of California, Berkeley, Berkeley, United States
- dMicrosoft (United States), Redmond, United States
- eStanford University, Stanford, United States
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Abstract
Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of computation, which on the one hand yields strong computational lower bounds for powerful classes of algorithms and on the other hand helps guide the development of efficient algorithms. In this paper, we study two of the most popular restricted computational models, the statistical query framework and low-degree polynomials, in the context of high-dimensional hypothesis testing. Our main result is that under mild conditions on the testing problem, the two classes of algorithms are essentially equivalent in power. As corollaries, we obtain new statistical query lower bounds for sparse PCA, tensor PCA and several variants of the planted clique problem.