Trend Filtering on Graphs

We introduce a family of adaptive estimators on graphs, based on penalizing the \ell_1 norm of discrete graph differences. This generalizes the idea of trend filtering (Kim et al., 2009, Tibshirani, 2014) used for univariate nonparametric regression, to graphs. Analogous to the univariate case, graph trend filtering exhibits a level of local adaptivity unmatched by the usual \ell_2-based graph smoothers. It is also defined by a convex minimization problem that is readily solved (e.g., by fast ADMM or Newton algorithms). We demonstrate the merits of graph trend filtering through examples and theory.