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Intertwining wavelets or Multiresolution analysis on graphs through random forests

Abstract : We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.
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https://hal.archives-ouvertes.fr/hal-01877023
Contributor : Fabienne Castell <>
Submitted on : Wednesday, September 19, 2018 - 11:09:18 AM
Last modification on : Wednesday, December 23, 2020 - 3:10:36 AM

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Fabienne Castell, Luca Avena, Alexandre Gaudilliere, Clothilde Melot. Intertwining wavelets or Multiresolution analysis on graphs through random forests. Applied and Computational Harmonic Analysis, Elsevier, In press, ⟨10.1016/j.acha.2018.09.006⟩. ⟨hal-01877023⟩

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