## Trend Filtering on Graphs

** Yu-Xiang Wang, James Sharpnack, Alexander J. Smola, Ryan J. Tibshirani**; 17(105):1−41, 2016.

### Abstract

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 both examples and theory.

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