Spectrum Estimation through Kirchhoff Random Forests

TL;DR

This paper introduces a Monte Carlo-based method using Kirchhoff random forests to efficiently estimate Laplacian eigenvalues of graphs, with potential applications to spectral analysis of symmetric matrices.

Contribution

It presents a novel spectral estimation technique leveraging Kirchhoff random forests, achieving linear sampling complexity for graph Laplacians and extending to symmetric matrices via double covers.

Findings

Sampling cost is linear in the number of nodes, up to a logarithmic factor.

Method effectively estimates Laplacian eigenvalues using spectral quantities along forest trajectories.

Applicable to spectral estimation of symmetric matrices through graph double covers.

Abstract

Given a non-oriented edge-weighted graph, we show how to make some estimation of the associated Laplacian eigenvalues through Monte Carlo evaluation of spectral quantities computed along Kirchhoff random rooted spanning forest trajectories. The sampling cost of this estimation is only linear in the node number, up to a logarithmic factor. By associating a double cover of such a graph with any symmetric real matrix, we can then perform spectral estimation in the same way for the latter.