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.
翻译:研究人员目前采用多种方法预测和证实高维统计估计问题的信息计算差距,一个突出的方法是确定限制性计算模式的局限性,一方面,这些限制计算模式为强力算法类别提供了强大的计算下限,另一方面,有助于指导高效算法的发展。在本文中,我们研究了两种最受欢迎的限制性计算模型,即统计查询框架和低度多级多级模型,这是在高维假设测试的背景下进行的。我们的主要结果是,在试验问题的温和条件下,两种类型的算法基本上具有同等的能量。作为卷轴,我们获得了稀有的五氯苯甲醚、高压五氯苯甲和多种人种问题的新的统计下限。