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Researchers currently use a number of approaches to predict and substantiateinformation-computation gaps in high-dimensional statistical estimationproblems. A prominent approach is to characterize the limits of restrictedmodels of computation, which on the one hand yields strong computational lowerbounds for powerful classes of algorithms and on the other hand helps guide thedevelopment of efficient algorithms. In this paper, we study two of the mostpopular restricted computational models, the statistical query framework andlow-degree polynomials, in the context of high-dimensional hypothesis testing.Our main result is that under mild conditions on the testing problem, the twoclasses of algorithms are essentially equivalent in power. As corollaries, weobtain new statistical query lower bounds for sparse PCA, tensor PCA andseveral variants of the planted clique problem.

Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent