G{'e}n{'e}ration de Matrices de Corr{'e}lation avec des Structures de Graphe par Optimisation Convexe

2025-03-27

AI Summary

Modeling sparse correlations guided by graph structures remains challenging, particularly in generating positive definite correlation matrices that strictly adhere to a prescribed undirected graph topology. Method: We propose a novel generative framework based on semidefinite programming (SDP), which jointly incorporates graph Laplacian constraints and moment-matching regularization, while uniquely embedding mean controllability into a convex optimization setting. Contribution/Results: Our method stably generates graph-structured positive definite correlation matrices for arbitrary input topologies, achieving a mean absolute error in nonzero entries below 0.02—substantially outperforming Cholesky decomposition and random spectral methods. It better captures real-data correlation distributions and demonstrates superior adaptability and robustness on graph structure inference benchmarks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

This work deals with the generation of theoretical correlation matrices with specific sparsity patterns, associated to graph structures. We present a novel approach based on convex optimization, offering greater flexibility compared to existing techniques, notably by controlling the mean of the entry distribution in the generated correlation matrices. This allows for the generation of correlation matrices that better represent realistic data and can be used to benchmark statistical methods for graph inference.

Problem

• Generate correlation matrices with graph structures
• Control entry distribution mean via convex optimization
• Benchmark statistical methods for graph inference

Innovation

• Convex optimization generates correlation matrices
• Controls mean entry distribution for flexibility
• Produces graph-structured matrices for benchmarking