Graph structure learning for stable processes

2026-01-09

AI Summary

This study addresses the challenge of modeling pathwise conditional independence structures and learning sparse graphical representations among stable processes with heterogeneous marginal stability indices. To this end, the authors propose the Ising–Hüsler–Reiss process, which introduces an Ising-type parametrization into the Hüsler–Reiss family of stable processes for the first time. In this framework, zeros in the precision matrix encode the conditional independence graph, while Ising weights capture the asymmetry of Lévy measure jumps across orthants. By integrating small-time Lévy process approximations, sparse graph estimation, and identification of zero patterns in the precision matrix, the method yields a consistent estimator for both the graph structure and the asymmetry parameters. Numerical simulations and an application to modeling dependencies in stock returns demonstrate the effectiveness of jointly learning sparse graphs and jump asymmetries.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We introduce Ising-H\"usler-Reiss processes, a new class of multivariate L\'evy processes that allows for sparse modeling of the path-wise conditional independence structure between marginal stable processes with different stability indices. The underlying conditional independence graph is encoded as zeroes in a suitable precision matrix. An Ising-type parametrization of the weights for each orthant of the L\'evy measure allows for data-driven modeling of asymmetry of the jumps while retaining an arbitrary sparse graph. We develop consistent estimators for the graphical structure and asymmetry parameters, relying on a new uniform small-time approximation for L\'evy processes. The methodology is illustrated in simulations and a real data application to modeling dependence of stock returns.

Problem

• graph structure learning
• stable processes
• conditional independence
• Lévy processes
• asymmetry

Innovation

• Ising-Hüsler-Reiss process
• graph structure learning
• stable processes
• Lévy measure asymmetry
• sparse precision matrix