Dimension Reduction in Multivariate Extremes via Latent Linear Factor Models

2026-02-26

AI Summary

This study addresses the challenge of modeling complex tail dependence structures in high-dimensional extreme-value data, where traditional methods often fail to balance interpretability, scalability, and the curse of dimensionality. The authors propose a novel extremal dependence model based on low-dimensional latent linear factors, incorporating explicit dimension reduction and sparse loading structures to simultaneously enhance model interpretability and computational efficiency. They establish theoretical identifiability conditions for the model parameters and develop a constructive recovery approach leveraging the marginal-free tail pairwise dependence matrix. Applied to a high-dimensional wind energy dataset, the method effectively estimates the joint extremal risk of multiple wind turbines simultaneously operating below cut-in wind speeds, demonstrating its scalability and practical utility in real-world scenarios.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose a new and interpretable class of high-dimensional tail dependence models based on latent linear factor structures. Specifically, extremal dependence of an observable vector is assumed to be driven by a lower-dimensional latent $K$-factor model, where $K \ll d$, thereby inducing an explicit form of dimension reduction. Geometrically, this is reflected in the support of the associated spectral dependence measure, whose intrinsic dimension is at most $K-1$. The loading structure may additionally exhibit sparsity, meaning that each component is influenced by only a small number of latent factors, which further enhances interpretability and scalability. Under mild structural assumptions, we establish identifiability of the model parameters and provide a constructive recovery procedure based on a margin-free tail pairwise dependence matrix, which also yields practical rank-based estimation methods. The framework combines naturally with marginal tail models and is particularly well suited to high-dimensional settings. We illustrate its applicability in a spatial wind energy application, where the latent factor structure enables tractable estimation of the risk that a large proportion of turbines simultaneously fall below their cut-in wind speed thresholds.

Problem

• dimension reduction
• multivariate extremes
• tail dependence
• latent factor model
• high-dimensional statistics

Innovation

• latent factor model
• dimension reduction
• multivariate extremes
• tail dependence
• spectral measure