Spectrum Estimation through Kirchhoff Random Forests

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.

💡 Research Summary

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The paper addresses the problem of estimating the spectral distribution of large symmetric matrices, focusing on graph Laplacians, without computing the full eigendecomposition. Exact diagonalisation of an n × n matrix costs O(n³) and is infeasible for modern data sets. In many applications, however, one only needs the cumulative distribution function (CDF) of eigenvalues, i.e., F(q)=σ(