Intertwining wavelets or Multiresolution analysis on graphs through random forests

Fabienne Castell (1) , Luca Avena (2) , Alexandre Gaudilliere (1) , Clothilde Melot (1)

1 I2M - Institut de Mathématiques de Marseille
2 Mathematical institute

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.

Domaines: Probabilités [math.PR]; Théorie de l'information et codage [math.IT]

Fabienne Castell, Luca Avena, Alexandre Gaudilliere, Clothilde Melot. Intertwining wavelets or Multiresolution analysis on graphs through random forests. Applied and Computational Harmonic Analysis, 2020, Applied Computational Harmonic Analysis, 48 (3), pp.949-992. ⟨10.1016/j.acha.2018.09.006⟩. ⟨hal-01877023⟩