Approximating M-matrix in Learning Directed Acyclic Graphs Using Methods Involve Semidefinite Matrix Constraints

Abstract

The task of deducing directed acyclic graphs from observational data has gained significant attention recently due to its broad applicability. Consequently, connecting the log-det characterization domain with the set of M-matrices defined over the cone of positive definite matrices has emerged as a crucial approach in this field. However, experimentally collected data often deviates from the expected positive semidefinite structure due to introduced noise, posing a challenge in maintaining its physical structure. In this paper, we address this challenge by proposing four methods to reconstruct the initial matrix while maintaining its physical structure. Leveraging advanced techniques, including sequential quadratic programming (SQP), we minimize the impact of noise, ensuring the recovery of the reconstructed matrix. We provide a rigorous proof of convergence for the SQP method, highlighting its effectiveness in achieving reliable reconstructions. Through comparative numerical analyses, we demonstrate the effectiveness of our methods in preserving the original structure of the initial matrix, even in the presence of noise.