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：图估计是网络分析中最基本的问题之一，过去十年间受到了广泛关注。从统计角度来看，Gao等人(2015)针对随机块模型（SBM）和非参数图估计建立了极小极大误差率。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呈指数增长。从计算角度来看，已知最佳的多项式时间估计器基于通用奇异值阈值法（USVT），但其估计误差率远低于极小极大值。自然会产生疑问：这种差距是否本质存在？USVT的计算最优性或图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。本研究首次迈向这一目标，通过低度多项式为图估计的计算障碍提供了严格证据。具体而言，在SBM和非参数图估计中，我们发现：在广泛参数范围内，低度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)最近发展的低度多项式理论，同时克服了将其应用于一般图估计问题的若干关键挑战。通过利用主要结果，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。