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Latent Gaussian and Hüsler--Reiss Graphical Models with Golazo Penalty

Ignacio Echave-Sustaeta Rodríguez, Frank Röttger

TL;DR

This work addresses latent-variable contamination in Gaussian and Hüsler--Reiss graphical models by introducing the Golazo penalty, a convex, flexible generalization that supports sparse-plus-low-rank decompositions and a range of structural constraints (e.g., adaptive sparsity, positivity, MTP2/EMTP2). An ADMM-based algorithm solves three-block optimization problems for both Gaussian and HR settings, including Laplacian-constrained variants, and is demonstrated on simulated data and real datasets (gene expression and flight delays). The results show that Golazo constraints can yield robust structure recovery and improved cross-validated likelihoods, particularly under total-positivity constraints, and the framework accommodates partial sparsity and prior knowledge. The proposed approach provides a unified, tunable toolkit for latent structure discovery in both standard and extreme-value graphical models, with promising avenues for theory, ensembles, and refitting strategies.

Abstract

The existence of latent variables in practical problems is common, for example when some variables are difficult or expensive to measure, or simply unknown. When latent variables are unaccounted for, structure learning for Gaussian graphical models can be blurred by additional correlation between the observed variables that is incurred by the latent variables. A standard approach for this problem is a latent version of the graphical lasso that splits the inverse covariance matrix into a sparse and a low-rank part that are penalized separately. This approach has recently been extended successfully to Hüsler--Reiss graphical models, which can be considered as an analogue of Gaussian graphical models in extreme value statistics. In this paper we propose a generalization of structure learning for Gaussian and Hüsler--Reiss graphical models via the flexible Golazo penalty. This allows us to introduce latent versions of for example the adaptive lasso, positive dependence constraints or predetermined sparsity patterns, and combinations of those. We develop algorithms for both latent graphical models with the Golazo penalty and demonstrate them on simulated and real data.

Latent Gaussian and Hüsler--Reiss Graphical Models with Golazo Penalty

TL;DR

This work addresses latent-variable contamination in Gaussian and Hüsler--Reiss graphical models by introducing the Golazo penalty, a convex, flexible generalization that supports sparse-plus-low-rank decompositions and a range of structural constraints (e.g., adaptive sparsity, positivity, MTP2/EMTP2). An ADMM-based algorithm solves three-block optimization problems for both Gaussian and HR settings, including Laplacian-constrained variants, and is demonstrated on simulated data and real datasets (gene expression and flight delays). The results show that Golazo constraints can yield robust structure recovery and improved cross-validated likelihoods, particularly under total-positivity constraints, and the framework accommodates partial sparsity and prior knowledge. The proposed approach provides a unified, tunable toolkit for latent structure discovery in both standard and extreme-value graphical models, with promising avenues for theory, ensembles, and refitting strategies.

Abstract

The existence of latent variables in practical problems is common, for example when some variables are difficult or expensive to measure, or simply unknown. When latent variables are unaccounted for, structure learning for Gaussian graphical models can be blurred by additional correlation between the observed variables that is incurred by the latent variables. A standard approach for this problem is a latent version of the graphical lasso that splits the inverse covariance matrix into a sparse and a low-rank part that are penalized separately. This approach has recently been extended successfully to Hüsler--Reiss graphical models, which can be considered as an analogue of Gaussian graphical models in extreme value statistics. In this paper we propose a generalization of structure learning for Gaussian and Hüsler--Reiss graphical models via the flexible Golazo penalty. This allows us to introduce latent versions of for example the adaptive lasso, positive dependence constraints or predetermined sparsity patterns, and combinations of those. We develop algorithms for both latent graphical models with the Golazo penalty and demonstrate them on simulated and real data.
Paper Structure (24 sections, 2 theorems, 39 equations, 15 figures, 2 algorithms)

This paper contains 24 sections, 2 theorems, 39 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Gaussian Graphical Model
  5. Multivariate Gaussians with Hidden Variables
  6. Laplacian-constrained Gaussian graphical model
  7. Multivariate extremes and the Hüsler--Reiss distribution
  8. Latent Hüsler--Reiss graphical models
  9. Golazo Constraints
  10. Learning latent Gaussian and Hüsler--Reiss graphical models via Golazo constraints
  11. Gaussian setting
  12. Learning Laplacian-constrained Gaussian Graphical Models under Golazo Constraints
  13. ADMM Algorithm
  14. Gaussian setting
  15. Hüsler--Reiss setting
  16. ...and 9 more sections

Key Result

Theorem 2.1

Chandrasekaran Let $A$ and $B$ denote the ground-truth sparse and low-rank components. Let and given a matrix $M$ and its tangent space $T(M)$, let Under the assumptions of Chandrasekaran, we have that the probability of having simultaneously is at least $1-2\exp(-|O|)$.

Figures (15)

  • Figure 1: Example of a Gaussian graphical model.
  • Figure 2: Graph with four observed variables and one hidden (left), and completely connected graph with four observed variables (right).
  • Figure 3: Two disconnected $4$-cycles with one hidden variable.
  • Figure 4: Results for the two cycles (red and black line become equal).
  • Figure 5: Average validation Hüsler--Reiss log-likelihood
  • ...and 10 more figures

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Theorem 2.1
  • Corollary 3.1
  • proof
  • Remark 1