Graphon estimation has been one of the most fundamental problems in network analysis and has received considerable attention in the past decade. From the statistical perspective, the minimax error rate of graphon estimation has been established by Gao et al (2015) for both stochastic block model (SBM) and nonparametric graphon estimation. The statistical optimal estimators are based on constrained least squares and have computational complexity exponential in the dimension. From the computational perspective, the best-known polynomial-time estimator is based on universal singular value thresholding (USVT), but it can only achieve a much slower estimation error rate than the minimax one. It is natural to wonder if such a gap is essential. The computational optimality of the USVT or the existence of a computational barrier in graphon estimation has been a long-standing open problem. In this work, we take the first step towards it and provide rigorous evidence for the computational barrier in graphon estimation via low-degree polynomials. Specifically, in both SBM and nonparametric graphon estimation, we show that for low-degree polynomial estimators, their estimation error rates cannot be significantly better than that of the USVT under a wide range of parameter regimes. Our results are proved based on the recent development of low-degree polynomials by Schramm and Wein (2022), while we overcome a few key challenges in applying it to the general graphon estimation problem. By leveraging our main results, we also provide a computational lower bound on the clustering error for community detection in SBM with a growing number of communities and this yields a new piece of evidence for the conjectured Kesten-Stigum threshold for efficient community recovery.

翻译：图论估计是网络分析中最基本的问题之一，在过去十年中受到了广泛关注。从统计学角度来看，Gao等人（2015）针对随机块模型（SBM）和非参数图论估计，确定了图论估计的极小化最大误差率。统计最优估计器基于约束最小二乘法，其计算复杂度随维度指数增长。从计算角度来看，最著名的多项式时间估计器基于通用奇异值阈值化（USVT），但其能达到的估计误差率远低于极小化最大误差率。自然要问：这种差距是本质性的吗？USVT的计算最优性或图论估计中是否存在计算障碍，一直是一个长期悬而未决的问题。在本工作中，我们迈出了解决这一问题的第一步，并通过低次多项式为图论估计中的计算障碍提供了严格证据。具体而言，在SBM和非参数图论估计中，我们证明：在广泛参数范围内，低次多项式估计器的估计误差率无法显著优于USVT。我们的结果基于Schramm和Wein（2022）近期对低次多项式的发展，同时克服了将其应用于一般图论估计问题时的若干关键挑战。借助主要结果，我们还为SBM中社区数量增长时的社区检测聚类误差提供了计算下界，这为高效社区恢复的Kesten-Stigum阈值猜想提供了新证据。