Sparse Graph Learning from Sparse Data via Fiedler Number Maximization

TL;DR

This paper introduces a novel method for learning sparse, connected graphs from limited data by maximizing the Fiedler number, improving robustness over previous algorithms.

Contribution

It develops greedy and parallel algorithms leveraging eigenvalue perturbation and Cheeger's inequality for robust sparse graph learning.

Findings

Fiedler number maximization enhances graph connectivity and robustness.

Proposed algorithms outperform previous sparse graph learning methods.

Simulation results validate the effectiveness of the approach.

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.