Minimax lower bounds for function estimation on graphs

In recent years there has been substantial interest in high-dimensional estimation and prediction problems on large graphs. These can in many cases be seen as high-dimensional or nonparametric regression or classification problems in which the goal is to learn a “smooth” function on a given graph. Various methods have been proposed to deal with such problems, motivated by a variety of applications. Any sensible method employs some form of regularisation that takes the geometry of the graph into account. Examples of methods that have been considered include penalised least squares regression using a Laplacianbased penalty (e.g. Ando and Zhang (2007); Belkin et al. (2004); Kolaczyk (2009); Smola and Kondor (2003); Zhu and Hastie (2005)), penalisation using the total variation norm (e.g. Sadhanala et al. (2016)) and Bayesian regularisa-