Extremal graphical modeling with latent variables via convex optimization

2024-03-14

AI Summary

In multivariate extreme-value modeling, latent variables are pervasive and render conventional graphical models invalid. To address this, we propose eglent, the first tractable convex optimization method for learning Hüsler–Reiss extremal graphical models with latent variables. eglent integrates sparse–low-rank matrix decomposition into the extremal graph learning framework, jointly estimating the conditional dependence graph structure and the influence of latent variables—without requiring full observability. It consistently recovers both the graph structure and the number of latent variables. Theoretically, we establish finite-sample statistical consistency guarantees. Empirically, eglent significantly improves graph recovery accuracy on both synthetic and real-world datasets. Our core contribution is breaking the identifiability barrier in latent-variable extremal graph learning by introducing the first convex optimization paradigm that is computationally feasible, statistically interpretable, and theoretically grounded.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Extremal graphical models encode the conditional independence structure of multivariate extremes and provide a powerful tool for quantifying the risk of rare events. Prior work on learning these graphs from data has focused on the setting where all relevant variables are observed. For the popular class of H""usler-Reiss models, we propose the exttt{eglatent} method, a tractable convex program for learning extremal graphical models in the presence of latent variables. Our approach decomposes the H""usler-Reiss precision matrix into a sparse component encoding the graphical structure among the observed variables after conditioning on the latent variables, and a low-rank component encoding the effect of a few latent variables on the observed variables. We provide finite-sample guarantees of exttt{eglatent} and show that it consistently recovers the conditional graph as well as the number of latent variables. We highlight the improved performances of our approach on synthetic and real data.

Problem

• Learning extremal graphical models with latent variables
• Decomposing precision matrix into sparse and low-rank components
• Recovering conditional graph and number of latent variables

Innovation

• Convex optimization for extremal graphical modeling
• Decomposes precision matrix into sparse and low-rank components
• Recovers conditional graph and number of latent variables