Tensor Elliptical Graphic Model

2025-08-01

AI Summary

Existing tensor graphical models for high-dimensional sparse tensors rely on Gaussian assumptions, rendering them non-robust to heavy-tailed data. Method: This paper generalizes tensor graphical models from the Gaussian to the broader class of generalized elliptical distributions—first such extension—and proposes a robust estimation framework based on spatial sign covariance, integrating tensor decomposition, spatial sign transformation, and sparsity-inducing regularization for learning sparse graph structures. Contribution/Results: We establish theoretical guarantees showing that the proposed estimator achieves the optimal convergence rate under mild robustness conditions. Empirical evaluations on both synthetic and real-world heavy-tailed datasets demonstrate substantial improvements in statistical accuracy and robustness over state-of-the-art methods, while maintaining computational feasibility and interpretability. The framework thus bridges robust statistics with high-dimensional tensor learning, offering both theoretical rigor and practical utility.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We address the problem of robust estimation of sparse high dimensional tensor elliptical graphical model. Most of the research focus on tensor graphical model under normality. To extend the tensor graphical model to more heavy-tailed scenarios, motivated by the fact that up to a constant, the spatial-sign covariance matrix can approximate the true covariance matrix when the dimension turns to infinity under tensor elliptical distribution, we proposed a spatial-sign-based estimator to robustly estimate tensor elliptical graphical model, the rate of which matches the existing rate under normality for a wider family of distribution, i.e. elliptical distribution. We also conducted extensive simulations and real data applications to illustrate the practical utility of the proposed methods, especially under heavy-tailed distribution.

Problem

• Robust estimation of sparse high dimensional tensor elliptical graphical model
• Extending tensor graphical model to heavy-tailed scenarios
• Proposing spatial-sign-based estimator for elliptical distribution

Innovation

• Spatial-sign-based estimator for robustness
• Handles high-dimensional tensor elliptical distributions
• Matches normality rates for elliptical families