Sparse Graph Learning from Sparse Data via Fiedler Number Maximization

2026-04-28

AI Summary

This work addresses the challenge of sparse graph learning under extreme underdetermined settings, where the number of observations is far smaller than the signal dimension and the underlying distribution is unknown. The authors propose a novel approach that incorporates graph connectivity as a regularizer by maximizing the Fiedler value—the second smallest eigenvalue of the graph Laplacian—a strategy introduced here for the first time in sparse graph learning to ensure both connectivity and robustness. Leveraging eigenvalue perturbation theory, they devise a globally greedy optimization scheme and further develop a parallel recursive graph partitioning algorithm grounded in the Cheeger inequality. Experimental results demonstrate that the proposed method significantly outperforms existing algorithms in preserving graph connectivity and accurately recovering graph structure.

Abstract

We aim to learn a sparse and connected graph from sparse data, where the number of observations K can be substantially smaller than the signal dimension N for signals x in R^N, and the underlying distribution is unknown. In this severely ill-posed setting, we incorporate Fiedler number (the second eigenvalue of the graph Laplacian matrix that quantifies connectedness) as a robust regularization term in the sparse graph learning objective. We first develop a greedy algorithm that iteratively selects one edge globally for weakening/removal to reduce the objective, leveraging eigenvalue perturbation theorems that bound the adverse effect of an edge change to the Fiedler number. Next, we design a parallel variant, based on the Cheeger's inequality, that recursively partitions an input graph into two sub-graphs using an approximate Cheeger cut to distributedly find an optimal edge. Simulation experiments show that Fiedler number maximization robustifies sparse graph estimates, outperforming previous sparse graph learning algorithms.

Problem

• sparse graph learning
• sparse data
• Fiedler number
• graph connectivity
• ill-posed problem

Innovation

• Fiedler number
• sparse graph learning
• graph Laplacian
• Cheeger cut
• eigenvalue perturbation