### 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 and fast algorithm and demonstrate its use with examples.

The examples include image analysis, a simulation applicable to discrete spatial variation, and classification. In our examples, penalized regression improves upon kernel smoothing in terms of identifying local extreme values on planar graphs. In all examples we use fully automatic procedures for setting the smoothing parameters. Supplemental materials are available online.

Translated title of the contribution | Regression on a graph |
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Original language | English |

Pages (from-to) | 432-447 |

Number of pages | 16 |

Journal | Journal of Computational and Graphical Statistics |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2011 |

### Bibliographical note

Publisher: American Statistical Association### Keywords

- active set algorithm
- image analysis
- penalized regression
- total variation

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## Cite this

*Journal of Computational and Graphical Statistics*,

*20*(2), 432-447. https://doi.org/10.1198/jcgs.2011.09203