Graph Tikhonov Regularization and Interpolation via Random Spanning Forests

TL;DR

This paper introduces Monte Carlo estimators based on random spanning forests for efficient Tikhonov regularization and interpolation on graphs, with theoretical analysis and practical hyperparameter tuning.

Contribution

It presents novel RSF-based estimators for graph regularization and interpolation, enabling faster computations and hyperparameter tuning in various algorithms.

Findings

Estimators have favorable mean and variance properties.

Methods are adaptable to multiple graph-based algorithms.

Experimental results show improved run time performance.

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. TR is a component in many well-known algorithms, and we show how the proposed estimators can be easily adapted to avoid expensive intermediate steps in generalized semi-supervised learning, label propagation, Newton's method and iteratively reweighted least squares. In the experiments, we illustrate the proposed methods on several problems and provide observations on their run time.