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. 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, including computer code for the examples, and proofs, are available online.
In graph interpolation or regression on a graph, one is given vectors at some of the vertices of a graph, and is tasked with inferring the vectors at the other (unknown) vertices REF .
Nonparametric Regression on a Graph
Journal of Computational and Graphical Statistics
Mathematics