Trend Filtering on Graphs

TL;DR

This paper extends trend filtering techniques from univariate regression to graph-structured data, introducing adaptive estimators that improve local fit and are computationally efficient.

Contribution

It generalizes trend filtering to graphs, providing a convex optimization framework with demonstrated theoretical and practical advantages.

Findings

Graph trend filtering is highly adaptive to local data features.

The method is computationally efficient with existing algorithms.

Theoretical analysis supports its effectiveness.

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 examples and theory.