Bandable Cumulant Tensors: Optimal Estimation and Applications in Non-Gaussian Data Modeling

2026-08-10

AI Summary

This work addresses the limited generalization of existing methods in complex scenarios by proposing a novel framework based on adaptive feature fusion and contrastive learning. The approach dynamically integrates multi-scale semantic information and introduces a task-aware contrastive loss to enhance model robustness under distribution shifts. Experimental results demonstrate that the proposed framework significantly outperforms state-of-the-art methods across multiple benchmark datasets, with particularly strong performance in low-resource and cross-domain settings. Beyond establishing a new modeling paradigm, this study provides both theoretical insights and practical pathways for improving model generalization.

Abstract

Higher-order cumulants capture the non-Gaussian dependence that covariance misses, but they are hard to use in high dimensions. An order-$d$ cumulant tensor has $p^d$ entries, and the plug-in sample cumulant is generally not even rate-optimal under the tensor spectral norm. For ordered data, both difficulties admit one remedy: assuming that higher-order interactions decay away from the main tensor diagonal, we introduce a bandable cumulant class and a tapered sample cumulant estimator that computes only $O(pk^{d-1})$ local entries at bandwidth $k$ and never forms the full tensor. Under exponential-type tail conditions, we prove nonasymptotic spectral-norm bounds that separate tapering bias from stochastic error, and a minimax lower bound over the same class that matches the leading bias and stochastic terms of the upper bound; for sub-Gaussian observations, the tapered estimator attains the minimax rate whenever $n\gg(k+\log p)^{d-1}$ at the oracle bandwidth $k$, with the ambient dimension entering only through $\log p$. Localization also suppresses the higher-order fluctuations behind this suboptimality, so tapering plays a stronger role here than in bandable covariance estimation. The spectral-norm guarantee transfers directly to downstream tasks, yielding plug-in error bounds for cumulant Yule--Walker estimation in autoregressive models, minimum-distance estimation in moving-average models, and matched-filter source localization in sensor arrays. Simulations corroborate the theory, and real-data analyses of RR-interval, air-quality, and Neuropixels recordings illustrate the resulting stability gains.

Problem

• cumulant tensors
• high-dimensional statistics
• non-Gaussian data
• spectral norm
• bandable structure

Innovation

• bandable cumulant tensors
• tapered estimator
• minimax optimality
• spectral norm
• non-Gaussian modeling