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Construction of Tight Frames on Graphs and Application to Denoising

  • Franziska Göbel
  • Gilles Blanchard
  • Ulrike von Luxburg
Chapter
Part of the Springer Handbooks of Computational Statistics book series (SHCS)

Abstract

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 is adapted to the underlying geometrical structure of the xi, has finitely many elements, and these elements are localized in frequency as well as in space. This construction follows the ideas of Hammond et al. (Appl Comput Harmon Anal 30:129–150, 2011), with the key point that we construct a tight (or Parseval) frame. This means we have a very simple, explicit reconstruction formula for every function f defined on the graph from the coefficients given by its scalar product with the frame elements. We use this representation in the setting of denoising where we are given noisy observations of a function f defined on the graph. By applying a thresholding method to the coefficients in the reconstruction formula, we define an estimate of f whose risk satisfies a tight oracle inequality.

Keywords

Neighborhood graph Tight frame Dictionary learning Denoising Thresholding Oracle inequality 

Notes

Acknowledgements

The authors acknowledge the financial support of the German DFG, under the Research Unit FOR-1735 “Structural Inference in Statistics—Adaptation and Efficiency.”

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Franziska Göbel
    • 1
  • Gilles Blanchard
    • 1
  • Ulrike von Luxburg
    • 2
  1. 1.Institute of MathematicsUniversity of PotsdamPotsdamGermany
  2. 2.Department of Computer ScienceUniversity of TübingenTübingenGermany

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