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.
Cite
@article{arxiv.1707.04616,
title = {Intertwining wavelets or Multiresolution analysis on graphs through random forests},
author = {Luca Avena and Fabienne Castell and Alexandre Gaudillière and Clothilde Mélot},
journal= {arXiv preprint arXiv:1707.04616},
year = {2018}
}
Comments
39 pages, 12 figures