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Extremal graphical modeling with latent variables via convex optimization

Sebastian Engelke, Armeen Taeb

TL;DR

This work addresses learning extremal graphical models in the presence of latent variables by introducing eglatent, a convex program that decomposes the marginal HR precision \tilde{\Theta} into a sparse observed-graph part and a low-rank latent-effect part. It establishes a Schur-complement-like relationship \tilde{\Theta} = \Theta_O - \Theta_{OH}\Theta_H^{-1}\Theta_{HO} and promotes a sparse-plus-low-rank structure to identify both the conditional graph among observed variables and the latent factors. The authors provide identifiability conditions and finite-sample consistency guarantees, derive an empirical variogram estimator, and demonstrate improved structure recovery and latent-variable counting on synthetic data and a real flight-delay dataset. The method yields more interpretable extremal models with better tail-dependency representations, enabling robust risk assessment for extreme events in high dimensions. The work also offers practical guidance and open-source tooling for practitioners analyzing extremes with latent confounders.

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 Hüsler-Reiss models, we propose the \texttt{eglatent} method, a tractable convex program for learning extremal graphical models in the presence of latent variables. Our approach decomposes the Hüsler-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 \texttt{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.

Extremal graphical modeling with latent variables via convex optimization

TL;DR

This work addresses learning extremal graphical models in the presence of latent variables by introducing eglatent, a convex program that decomposes the marginal HR precision \tilde{\Theta} into a sparse observed-graph part and a low-rank latent-effect part. It establishes a Schur-complement-like relationship \tilde{\Theta} = \Theta_O - \Theta_{OH}\Theta_H^{-1}\Theta_{HO} and promotes a sparse-plus-low-rank structure to identify both the conditional graph among observed variables and the latent factors. The authors provide identifiability conditions and finite-sample consistency guarantees, derive an empirical variogram estimator, and demonstrate improved structure recovery and latent-variable counting on synthetic data and a real flight-delay dataset. The method yields more interpretable extremal models with better tail-dependency representations, enabling robust risk assessment for extreme events in high dimensions. The work also offers practical guidance and open-source tooling for practitioners analyzing extremes with latent confounders.

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 Hüsler-Reiss models, we propose the \texttt{eglatent} method, a tractable convex program for learning extremal graphical models in the presence of latent variables. Our approach decomposes the Hüsler-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 \texttt{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.
Paper Structure (48 sections, 31 theorems, 152 equations, 11 figures, 2 tables)

This paper contains 48 sections, 31 theorems, 152 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Notation
  4. Background
  5. Multivariate extreme value theory
  6. Extremal graphical models
  7. Latent Hüsler--Reiss models and the eglatent method
  8. Latent Hüsler--Reiss models
  9. Sparse plus low-rank decomposition
  10. Inference for latent Hüsler--Reiss graphical models
  11. Empirical extremal variogram matrix
  12. Parameter estimation and structure learning
  13. Consistency guarantees for eglatent
  14. Technical setup
  15. Conditions on the Hessian $\mathbb{I}^\star$
  16. ...and 33 more sections

Key Result

Proposition 1

Let $Y \in \mathbb{R}^d$ follow a Hüsler--Reiss distribution with precision matrix $\Theta$. Then,

Figures (11)

  • Figure 1: One-factor graph with one latent variable with four observed variables $O_1,\dots, O_4$ and one latent variable $H$ (left) and its marginalization on the observed variables (right).
  • Figure 2: Left: $F$-score of our proposed method eglatent (solid line) and eglearn (dashed line) as function of the regularization parameter with larger $F$-scores being better; top axis shows the number of estimated latent variables. Right: the likelihood of the same methods evaluated on a validation data set; the top axis shows the number of estimated edges in the latent model.
  • Figure 3: $F$-score (top row) and estimated number of latent variables (middle row) of eglatent method with the selection of the tuning parameter based on the oracle and validation on the $F$-score for the cycle graph with $h=1,2,3$ latent variables and different effective sample sizes $k=200,1000,5000$. The bottom row shows the difference between best eglatent and best eglearn log-likelihoods on the validation set.
  • Figure 4: Left: $F$-score of eglatent for different regularization parameters $\gamma \in \{1,4,8,20\}$ and eglearn; top axis shows the average number of estimated latent variables in eglatent. Right: the log-likelihood of the same methods evaluated on a validation data set; the top axis shows the average number of estimated edges in each model.
  • Figure 5: Airports in the Southern U.S. (dots) and flight connections, where the thickness of the nodes indicates the average number of daily flights at the airports. Left: flight connection graph with an edge between any pair of airports with daily flights. Center: estimated graph of optimal eglearn model. Right: estimated sub-graph corresponding to observed variables of optimal eglatent model.
  • ...and 6 more figures

Theorems & Definitions (65)

  • Definition 1
  • Definition 2
  • Proposition 1: Lemma 1 and Proposition 3 of engelke2020
  • Definition 3: Latent Hüsler--Reiss models
  • Theorem 2
  • Remark 1
  • Remark 2: Dependency on $h$ and graph structure
  • Remark 3: Choice of $\gamma$
  • Theorem 3
  • Remark 4
  • ...and 55 more