Intertwining wavelets or Multiresolution analysis on graphs through random forests

TL;DR

This paper introduces a multiscale analysis method for functions on graph vertices using random forests and intertwining relations, resulting in a wavelet basis with tunable parameters for localization.

Contribution

It presents a novel graph wavelet construction based on random spanning forests and Markov intertwining, with explicit formulas and parameter tuning guidelines.

Findings

Provides explicit reconstruction formula and bounds.

Demonstrates effectiveness through numerical experiments.

Offers parameter selection strategy for localization balance.

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.