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Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent
arXiv - CS - Computational Complexity Pub Date : 2020-09-13 , DOI: arxiv-2009.06107
Matthew Brennan and Guy Bresler and Samuel B. Hopkins and Jerry Li and Tselil Schramm

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.

中文翻译:

统计查询算法和低度测试几乎等价

研究人员目前使用多种方法来预测和证实高维统计估计问题中的信息计算差距。一个突出的方法是描述受限计算模型的限制,一方面为强大的算法类别产生强大的计算下界,另一方面有助于指导高效算法的开发。在本文中,我们在高维假设检验的背景下研究了两种最流行的受限计算模型,即统计查询框架和低次多项式。我们的主要结果是,在测试问题的温和条件下,两类算法的功效基本相同。作为推论,我们获得了稀疏 PCA 的新统计查询下界,
更新日期:2020-11-12
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