We study the optimal estimation of probability matrices of random graph models generated from graphons. This problem has been extensively studied in the case of step-graphons and H\"older smooth graphons. In this work, we characterize the regularity of graphons based on the decay rates of their eigenvalues. Our results show that for such classes of graphons, the minimax upper bound is achieved by a spectral thresholding algorithm and matches an information-theoretic lower bound up to a log factor. We provide insights on potential sources of this extra logarithm factor and discuss scenarios where exactly matching bounds can be obtained. This marks a difference from the step-graphon and H\"older smooth settings, because in those settings, there is a known computational-statistical gap where no polynomial time algorithm can achieve the statistical minimax rate. This contrast reflects a deeper observation that the spectral decay is an intrinsic feature of a graphon while smoothness is not.

翻译：我们研究了由图函数生成的随机图模型概率矩阵的最优估计问题。该问题在阶梯图函数和Hölder光滑图函数情形下已得到广泛研究。本文中，我们依据图函数特征值的衰减率来刻画其正则性。我们的结果表明，对于此类图函数，极小极大上界可通过谱阈值算法达到，并且与信息论下界仅相差一个对数因子。我们深入探讨了这一额外对数因子的潜在来源，并讨论了能够获得精确匹配界的情形。这与阶梯图函数和Hölder光滑设定形成对比，因为在后两种设定中，已知存在计算-统计间隙，即不存在多项式时间算法能达到统计极小极大速率。这一差异反映了一个更深刻的观察：谱衰减是图函数的内在特性，而光滑性则不然。