Minimax Optimal Probability Matrix Estimation For Graphon With Spectral Decay

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.