Regression on a Graph

TL;DR

This paper extends nonparametric regression to graph-structured data, introducing a total-variation penalization method with a new fast algorithm, demonstrating improved estimation in spectroscopic and spatial data examples.

Contribution

It develops a graph-based total-variation penalized regression method with an efficient algorithm, expanding the applicability of penalized regression to complex graph structures.

Findings

Penalized regression outperforms kernel smoothing in detecting local extrema.

The new algorithm efficiently solves the graph-based regression problem.

Automatic smoothing parameter selection is effective across examples.

Abstract

The `Signal plus Noise' model for nonparametric regression can be extended to the case of observations taken at the vertices of a graph. This model includes many familiar regression problems. This article discusses the use of the edges of a graph to measure roughness in penalized regression. Distance between estimate and observation is measured at every vertex in the $L_2$ norm, and roughness is penalized on every edge in the $L_1$ norm. Thus the ideas of total-variation penalization can be extended to a graph. The resulting minimization problem presents special computational challenges, so we describe a new, fast algorithm and demonstrate its use with examples. Further examples include a graphical approach that gives an improved estimate of the baseline in spectroscopic analysis, and a simulation applicable to discrete spatial variation. In our example, penalized regression outperforms kernel smoothing in terms of identifying local extreme values. In all examples we use fully automatic procedures for setting the smoothing parameters.