Tensor Covariance Estimation via Kronecker-Structured Sparse Inverse Cholesky

2026-08-14

AI Summary

This study addresses the curse of dimensionality in high-dimensional tensor covariance estimation by proposing a unified framework based on Kronecker-structured Sparse Inverse Cholesky (KSIC) projection. Integrating nonparametric and parametric estimation mechanisms, the method achieves geometry-aware representation through information geometric projection, nested Kullback-Leibler divergence minimization, and moment matching, while leveraging cross-modal information to enhance robustness in small-sample regimes. Experimental results demonstrate that this framework achieves state-of-the-art accuracy and scalability in high-dimensional, low-sample settings. Furthermore, its successful application to spatiotemporal temperature anomaly detection and fMRI data analysis validates its effectiveness in precisely modeling complex tensor data, thereby overcoming critical challenges in high-dimensional statistical inference.

Abstract

High-dimensional multi-way (tensor) data pose significant challenges for covariance estimation due to the curse of dimensionality. We introduce a unified framework for scalable estimation of tensor covariances based on a Kronecker-structured sparse inverse Cholesky (KSIC) projection. Our approach is grounded in the geometry of information projection, defining the estimator as the moment-matching projection of a target distribution onto a manifold characterized by sparse, Kronecker-factored inverse Cholesky factors. By leveraging physical or data-driven nearest-neighbor sparsity, KSIC provides a geometry-aware representation that is both statistically interpretable and computationally efficient. Our framework integrates two estimation regimes: a nonparametric estimator that projects the empirical covariance directly onto the manifold, utilizing the KSIC structure to implicitly regularize rank-deficient data; and a parametric estimator that fits generative covariance models (e.g., Matérn) by maximizing the likelihood of their KSIC projections, formulated as a nested double forward Kullback-Leibler minimization. Theoretically, we establish the conditions for the existence of the KSIC projection and finite-sample concentration rates for the nonparametric regime, proving that the KSIC estimator gainfully exploits cross-mode information and is robust to data scarcity. Numerical experiments demonstrate that the proposed KSIC estimators achieve state-of-the-art accuracy and scalability, particularly in settings with high dimensionality and limited sample sizes. We apply KSIC to spatiotemporal temperature anomalies and functional MRI data, demonstrating its broad applicability across diverse multi-way data domains.

Problem

• Tensor Covariance Estimation
• High-dimensional Data
• Curse of Dimensionality
• Small Sample Size
• Scalability

Innovation

• Tensor Covariance Estimation
• Kronecker-Structured Sparse Inverse Cholesky
• Information Projection
• High-dimensional Multi-way Data
• Nested Double Forward KL Minimization