Graph-Based Correlation Matrix Generation: A Convex Optimization Approach

2026-07-24

AI Summary

This study addresses the problem of generating theoretical correlation matrices that adhere to a prescribed sparsity pattern derived from a given graph structure, have unit diagonal entries, and allow for controllable mean values of off-diagonal elements. To this end, the authors propose a convex optimization–based construction method that projects an initial matrix onto an ellipsoid while enforcing a positive semidefinite constraint. This approach strictly preserves the target sparsity pattern and enables flexible tuning of the overall correlation strength. The method is theoretically guaranteed to yield feasible solutions, and simulations confirm that the generated matrices closely match the desired graph topology. Benchmark evaluations on real-world datasets from neuroscience and finance demonstrate its significant superiority over existing approaches such as GANs, offering a reliable and adjustable tool for correlation matrix generation in graphical model inference.

Abstract

This work addresses the generation of theoretical correlation matrices with prescribed sparsity patterns associated to graph structures. We propose a novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint. Several numerical schemes are implemented and compared. The problem falls within the broader class of matrix completion, where off-diagonal entries corresponding to absent edges are fixed to zero and diagonal entries are fixed to one. Beyond this structural constraint, the approach offers greater flexibility than existing methods by allowing control over the mean of the off-diagonal entry distribution, enabling the generation of correlation matrices that better reflect realistic data. This procedure is not designed to yield a uniform distribution over the feasible set; rather, it provides a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference. Theoretical guarantees on the existence of solutions are established, both in the general setting and under the additional mean constraint. Simulation studies illustrate the properties of the generated matrices with respect to graph structure. The methodology is applied to two real-world datasets from neuroscience and finance, and a comparison with GAN-based correlation matrix generation is provided.

Problem

• correlation matrix generation
• graph structure
• sparsity pattern
• matrix completion
• positive semidefiniteness

Innovation

• convex optimization
• correlation matrix generation
• graph-based sparsity
• matrix completion
• elliptope projection