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Given a neighborhood graph representation of a finite set of points$x_i\in\mathbb{R}^d,i=1,\ldots,n,$ we construct a frame (redundant dictionary)for the space of real-valued functions defined on the graph. This frame isadapted to the underlying geometrical structure of the $x_i$, has finitely manyelements, and these elements are localized in frequency as well as in space.This construction follows the ideas of Hammond et al. (2011), with the keypoint that we construct a tight (or Parseval) frame. This means we have a verysimple, explicit reconstruction formula for every function $f$ defined on thegraph from the coefficients given by its scalar product with the frameelements. We use this representation in the setting of denoising where we aregiven noisy observations of a function $f$ defined on the graph. By applying athresholding method to the coefficients in the reconstruction formula, wedefine an estimate of $f$ whose risk satisfies a tight oracle inequality.

Construction of Tight Frames on Graphs and Application to Denoising