GRACGE: Graph Signal Clustering and Multiple Graph Estimation

Abstract

In graph signal processing (GSP), complex datasets arise from several underlying graphs and in the presence of heterogeneity. Graph learning from heterogeneous graph signals often results in challenging high-dimensional multiple graph estimation problems, and prior information regarding which graph the data was observed is typically unknown. To address the above challenges, we develop a novel framework called GRACGE ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GRAph signal Clustering and multiple Graph Estimation</i> ) to partition the graph signals into clusters and jointly learn the multiple underlying graphs for each of the clusters. GRACGE advocates a regularized EM (rEM) algorithm where a structure fusion penalty with adaptive regularization parameters is imposed on the M-step. Such a penalty can exploit the structural similarities among graphs to overcome the curse of dimensionality. Moreover, we provide a non-asymptotic bound on the estimation error of the GRACGE algorithm, which establishes its computational and statistical guarantees. Furthermore, this theoretical analysis motivates us to adaptively re-weight the regularization parameters. With the adaptive regularization scheme, the final estimates of GRACGE will geometrically converge to the true parameters within statistical precision. Finally, experimental results on both synthetic and real data demonstrate the performance of the proposed GRACGE algorithm.