The minimax learning rates of normal and ising undirected graphical models

Abstract

Let G be an undirected graph with m edges and d vertices. We show that d-dimensional Ising models on G can be learned from n i.i.d. samples√within expected total variation distance some constant factor of min(formula presented) and that this rate is optimal. We show that the same rate holds for the class of d-dimensional multivariate normal undirected graphical√ models with respect to G. We also identify the optimal rate of min(formula presented) for Ising models with no external magnetic field.