Graph Tikhonov Regularization and Interpolation via Random Spanning Forests

Abstract

Novel Monte Carlo estimators are proposed to solve both the Tikhonov regularization (TR) and the interpolation problems on graphs. These estimators are based on random spanning forests (RSF), the theoretical properties of which enable to analyze the estimators' theoretical mean and variance. We also show how to perform hyperparameter tuning for these RSF-based estimators. Finally, TR or interpolation being a building block of several algorithms, we show how the proposed estimators can be easily adapted to avoid expensive intermediate steps in wellknown algorithms such as generalized semi-supervised learning, label propagation, Newton's method and iteratively reweighted least square. In the experiments, we illustrate the proposed methods on several problems and provide observations on their run time, which are comparable with the state-of-the-art.

Loop-erased partitioning of a network: monotonicities & analysis of cycle-free graphs

“…Further, these roots turn out to be well-distributed in the given network [4, Thm.1] and, conditional on the induced partition, their law is as the stationary measures of the random walk X restricted to each block [4, Prop.2.3] of the underlying partition. These and other features of the LEP have been recently exploited to build novel algorithms for the following different applications in data science: wavelets basis and filters for signal processing on graphs [3,33,34], estimate traces of discrete Laplacians and other diagonally dominant matrices [8], network renormalization [1,2], centrality measures [16] and statistical learning [7]. These applications give further motivations to explore in more details this LEP.…”