Smoothing Graph Signals via Random Spanning Forests

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 RSFbased 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.

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“…A remark. Reducing these results to the constant q case (Q = qI), one recovers the preliminary results presented in [25].…”

Section: Propositionsupporting

confidence: 80%

“…Our contributions. In the conference paper [25], we have already presented some preliminary results. Differing from [25], in this work,…”

Section: Introductionmentioning

confidence: 99%

Graph Tikhonov Regularization and Interpolation Via Random Spanning Forests

Pilavc

Amblard

Barthelmé

et al. 2021

IEEE Trans. on Signal and Inf. Process. over Networks

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show abstract

“…A remark. Reducing these results to the constant q case (Q = qI), one recovers the preliminary results presented in [24].…”

Section: Propositionsupporting

confidence: 80%

“…• We show how versatile these estimators are by adapting them to several graph-based problems such as generalized semi-supervised learning, label propagation, Newton's method and Iteratively Reweighted Least Squares (IRLS). A preliminary version of some of these results can be found in [24]. The Julia code to reproduce this paper's results is available on the authors' website.…”

Section: Our Contributions In This Workmentioning

confidence: 99%

Graph Tikhonov Regularization and Interpolation via Random Spanning Forests

Pilavci,

Amblard,

Barthelme

et al. 2020

Preprint

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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.

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“…Recently, a number of algorithms based on these results have been proposed as tools for tackling different problems in data science. These applications include: wavelets basis and filters for signal processing on graphs [3,29,28], trace estimation [10], network renormalization [4,5], centrality measures [16] and statistical learning [9]. The fact that random rooted forests proved a powerful tool in such different applied areas, was one of the staring motivation for the analysis of the partition measure in (1.9).…”

Section: Applications To Network Analysismentioning

confidence: 99%

Loop-erased partitioning of a graph: mean-field analysis

Avena

Gaudillière

Milanesi

et al. 2022

Electron. J. Probab.

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Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter q > 0. The related random trees-identifying the blocks of the partition-tend to cluster nodes visited by the random walk on time scale 1/q.We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for "densely connected nodes" and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of "giant" blocks.

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Smoothing Graph Signals via Random Spanning Forests

Cited by 7 publications

References 12 publications

Graph Tikhonov Regularization and Interpolation Via Random Spanning Forests

Graph Tikhonov Regularization and Interpolation Via Random Spanning Forests

Graph Tikhonov Regularization and Interpolation via Random Spanning Forests

Loop-erased partitioning of a graph: mean-field analysis

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