arXiv subjects

Ignacio Echave-Sustaeta Rodríguez

Publications and source records attributed to Ignacio Echave-Sustaeta Rodríguez.

4 recordsLinked to original sources

arXiv2026

Spectral Sparsification of Laplacian-Constrained Gaussian and Hüsler-Reiss Graphical Models

Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the Hüsler-Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well by running simulations on Erdős-Rényi and stochastic block model graphs, and we also showcase their applications to real data.

stat.ME↗

arXiv2026

Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification

Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.

stat.ME↗

arXiv2026

Extremal conditional independence for Hüsler-Reiss distributions via modular functions

We study extremal conditional independence for Hüsler-Reiss distributions, which is a parametric subclass of multivariate Pareto distributions. As the main contribution, we introduce two set functions, i.e.~functions which assign a value to the distribution and each of its marginals, and show that extremal conditional independence statements can be characterized by modularity relations for these functions. For the first function, we make use of the close connection between Hüsler-Reiss and Gaussian models to introduce a multiinformation-inspired measure $m^{\text{HR}}$ for Hüsler-Reiss distributions. For the second function, we consider an invariant $σ^2$ that is naturally associated to the Hüsler-Reiss parameterization and establish the second modularity criterion under additional positivity constraints. Together, these results provide new tools for describing extremal dependence structures in high-dimensional extreme value statistics. In addition, we study the geometry of a bounded subset of Hüsler-Reiss parameters and its relation with the Gaussian elliptope.

math.ST↗

arXiv2025

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

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.

stat.ME↗