Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning

2025-06-12

AI Summary

Learning undirected graph topologies from nodal observations in high-dimensional settings is computationally prohibitive for conventional methods, primarily due to the cubic complexity of estimating dense conditional correlation matrices. Method: This paper proposes a scalable low-rank graph learning framework: it is the first to model the conditional correlation matrix as intrinsically low-rank and formulates a novel GLasso variant incorporating explicit low-rank constraints; a differentiable Riemannian optimization algorithm is designed to jointly enforce sparsity and low-rank structure. Contribution/Results: Experiments on synthetic and real-world datasets demonstrate substantial improvements in the dimension–performance trade-off: the method reduces computational cost by an order of magnitude at the 1,000-dimensional scale while maintaining high-fidelity topology recovery. It establishes a theoretically grounded, scalable paradigm for large-scale graph signal processing.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

This paper addresses the problem of learning an undirected graph from data gathered at each nodes. Within the graph signal processing framework, the topology of such graph can be linked to the support of the conditional correlation matrix of the data. The corresponding graph learning problem then scales to the squares of the number of variables (nodes), which is usually problematic at large dimension. To tackle this issue, we propose a graph learning framework that leverages a low-rank factorization of the conditional correlation matrix. In order to solve for the resulting optimization problems, we derive tools required to apply Riemannian optimization techniques for this particular structure. The proposal is then particularized to a low-rank constrained counterpart of the GLasso algorithm, i.e., the penalized maximum likelihood estimation of a Gaussian graphical model. Experiments on synthetic and real data evidence that a very efficient dimension-versus-performance trade-off can be achieved with this approach.

Problem

• Learning undirected graphs from node data efficiently
• Reducing dimensionality via low-rank correlation factorization
• Optimizing performance in large-scale Gaussian graphical models

Innovation

• Low-rank factorization of conditional correlation matrices
• Riemannian optimization techniques for graph learning
• Low-rank constrained GLasso algorithm