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arXiv · 1707.04616

Intertwining wavelets or Multiresolution analysis on graphs through random forests

Luca Avena·Fabienne Castell·Alexandre Gaudillière·Clothilde Mélot
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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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Luca Avena, Fabienne Castell, Alexandre Gaudillière, Clothilde Mélot. 2017-07-14. Intertwining wavelets or Multiresolution analysis on graphs through random forests. https://arxiv.org/abs/1707.04616

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