We study a Bayesian approach to estimating a smooth function in the context of regression or classification problems on large graphs. We derive theoretical results that show how asymptotically optimal Bayesian regularisation can be achieved under an asymptotic shape assumption on the underlying graph and a smoothness condition on the target function, both formulated in terms of the graph Laplacian. The priors we study are randomly scaled Gaussians with precision operators involving the Laplacian of the graph.

We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that satisfy an asymptotic shape assumption and with a smoothness condition on the target function, both formulated in terms of the graph Laplacian.

No abstract

Abstract:We find precise small deviation asymptotics with respect to the Hilbert norm for some special Gaussian processes connected to two regression schemes studied by MacNeill and his coauthors. In addition, we also obtain precise small deviation asymptotics for the detrended Brownian motion and detrended Slepian process. MSC:60G15, 60J65, 62J05