Localization Estimator for High Dimensional Tensor Covariance Matrices

2026-01-11

AI Summary

This work addresses the challenges of the curse of dimensionality and complex dependence structures in covariance estimation for high-dimensional tensor data. The authors propose a class of multi-banded covariance models tailored to multi-way lattice structures and general decay patterns, together with a regularization framework based on localization functions. By integrating covariance tapering, high-dimensional statistical inference, and tensor modeling, the proposed estimator achieves minimax-optimal convergence rates under both the spectral and Frobenius norms. Theoretical analysis and numerical experiments—including an application to oceanic eddy data—demonstrate the method’s effectiveness and superiority over existing approaches.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

This paper considers covariance matrix estimation of tensor data under high dimensionality. A multi-bandable covariance class is established to accommodate the need for complex covariance structures of multi-layer lattices and general covariance decay patterns. We propose a high dimensional covariance localization estimator for tensor data, which regulates the sample covariance matrix through a localization function. The statistical properties of the proposed estimator are studied by deriving the minimax rates of convergence under the spectral and the Frobenius norms. Numerical experiments and real data analysis on ocean eddy data are carried out to illustrate the utility of the proposed method in practice.

Problem

• covariance matrix estimation
• tensor data
• high dimensionality
• localization
• multi-bandable covariance

Innovation

• tensor covariance
• localization estimator
• high-dimensional statistics
• minimax convergence rate
• multi-bandable structure