Optimization Algorithms for Graph Laplacian Estimation via ADMM and MM

In this paper, we study the graph Laplacian estimation problem under a given connectivity topology. We aim at enriching the unified graph learning framework proposed by Egilmez et al. and improve the optimality performance of the combinatorial graph Laplacian (CGL) case. We apply the well-known alternating direction method of multipliers (ADMM) and majorization–minimization (MM) algorithmic frameworks and propose two algorithms, namely, GLE-ADMM and GLE-MM, for graph Laplacian estimation. Both algorithms can achieve an optimality gap as low as $10^{-4}$, around three orders of magnitude more accurate than the benchmark. In addition, we find that GLE-ADMM is more computationally efficient in a dense topology (e.g., an almost complete graph), while GLE-MM is more suitable for sparse graphs (e.g., trees). Furthermore, we consider exploiting the leading eigenvectors of the sample covariance matrix as a nominal eigensubspace and propose a third algorithm, named GLENE, which is also based on ADMM. Numerical experiments show that the inclusion of a nominal eigensubspace significantly improves the estimation of the graph Laplacian, which is more evident when the sample size is smaller than or comparable to the problem dimension.