基于克罗内克结构稀疏逆乔列斯基的张量协方差估计
Tensor Covariance Estimation via Kronecker-Structured Sparse Inverse Cholesky
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中文总结 AI 辅助
针对高维张量数据协方差估计的维度灾难问题，提出基于克罗内克结构稀疏逆乔列斯基的统一估计框架，兼具统计可解释性与计算效率，在时空温度异常等多类数据上实现最优精度与可扩展性。
中文摘要 AI 辅助
高维多向（张量）数据因维度灾难给协方差估计带来重大挑战。我们引入了一个基于克罗内克结构稀疏逆乔列斯基（KSIC）投影的可扩展张量协方差估计统一框架。该方法以信息投影的几何为基础，将估计器定义为目标分布到由稀疏、克罗内克分解的逆乔列斯基因子构成的流形上的矩匹配投影。通过利用物理或数据驱动的最近邻稀疏性，KSIC提供了一种感知几何的表示，兼具统计可解释性与计算效率。该框架整合了两种估计模式：一种是非参数估计器，它将经验协方差直接投影到流形上，利用KSIC结构隐式地对秩亏数据进行正则化；另一种是参数估计器，它通过最大化生成协方差模型（如Matérn模型）的KSIC投影的似然来拟合这些模型，该过程被表述为嵌套的双重前向Kullback-Leibler最小化。理论上，我们确立了KSIC投影存在的条件以及非参数模式下的有限样本收敛速率，证明KSIC估计器可有效利用跨模式信息，且对数据稀缺具有鲁棒性。数值实验表明，所提出的KSIC估计器在高维且样本量有限的场景中达到了当前最优的精度与可扩展性。我们将KSIC应用于时空温度异常数据和功能性磁共振成像数据，证明其在不同多向数据领域具有广泛适用性。
英文摘要
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.
