Adaptive nonparametric regression with the K-nearest neighbour fused lasso

The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the Formula: see text-nearest-neighbours fused lasso, involves computing the Formula: see text-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the Formula: see text-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an Formula: see text-graph rather than a Formula: see text-nearest-neighbours graph and contrast it with the Formula: see text-nearest-neighbours fused lasso.