Graph Learning from Data under Structural and Laplacian Constraints

TL;DR

This paper introduces a new framework for learning graph structures from data by estimating graph Laplacian matrices under structural constraints, with probabilistic interpretations and specialized algorithms that outperform existing methods.

Contribution

It formulates graph learning as MAP estimation of GMRF models with Laplacian precision matrices, providing novel algorithms that incorporate structural constraints.

Findings

Algorithms outperform state-of-the-art in accuracy

Algorithms are more computationally efficient

Framework effectively incorporates structural constraints

Abstract

Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretations and (iii) associated algorithms. Specifically, graph learning problems are posed as estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori (MAP) parameter estimation of Gaussian-Markov random field (GMRF) models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.