Localization Estimator for High Dimensional Tensor Covariance Matrices

TL;DR

This work tackles high-dimensional covariance estimation for tensor data organized on a d-order lattice by introducing a flexible multi-bandable covariance class and a localization-based estimator that regularizes the sample covariance using a d-dimensional localization function. The authors derive minimax convergence rates under both spectral and Frobenius norms and prove these rates are optimal, extending banding and separable approaches to general tensor settings with nonseparable decay. The methodology is validated theoretically (through upper and lower bounds) and empirically via simulations and a case study on ocean eddy salinity tensors, where the localization estimator yields improved covariance estimates and downstream data assimilation performance. The framework accommodates irregular lattices and enables inverse covariance estimation, highlighting its practical impact for complex, multi-dimensional scientific data such as oceanography.

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.

Figures (7)

• Figure 1: (a) Sea level anomaly on September 1st, 2024 that displays an ocean eddy at 32$^\circ$N-35$^\circ$N and 158$^\circ$E-161$^\circ$E. (left) and the trivariate salinity field in the practical salinity unit (psu) in $10$ out of $35$ total layers and the average salinity with respect to the depth (right); (b) the sample correlation matrix of daily salinity changes from July 20th to September 12th 2024 at $47915$ grids with the $1/12^\circ\times 1/12^\circ$ spatial resolution and $35$ layers in depth (left), and at the $1369$ grids of the first layer at $0.5\mathrm{m}$ depth (right).
• Figure 2: Illustrations of the preserved $\mathbf{k}$-zones $\mathcal{H}_d(k)$ for univariate lattice $\mathcal{S}_1(10)$ ($d=1$ and $k=10$) (left), the $2$-order lattice $\mathcal{S}_2((10,10))$ (middle); and the $3$-order lattice $\mathcal{S}_3((10,10,10))$ (right). The entries that are filled with red or white represent $\boldsymbol \delta_{ij}\in\mathcal{H}_d(\mathbf{k})$ or $\boldsymbol \delta_{ij}\notin\mathcal{H}_d(\mathbf{k})$, respectively.
• Figure 3: The covariance decay function $\tau(\mathbf{k})$ for $d=2$ with $\tau(\mathbf{k})=k_1^{-0.5}+k_2^{-0.3}$ (a) and $\tau(\mathbf{k})=1.1^{-k_1}+1.2^{-k_2}$ (b).
• Figure 4: Four examples of the localization function $h(\mathbf{z})$ with $h$ being the multiplicative banding function $\prod_{\ell=1}^2\mathbb{I}\{z_\ell<1\}$ (a), the multiplicative tapering function $\prod_{\ell=1}^2\varphi(z_\ell;0.5,1)$ (b), the GC function $\text{GC}(2z)$ for $d=1$ (c) and $\text{GC}(2\sqrt{z_1^2+z_2^2})$ for $d=2$ (d).
• Figure 5: Heatmap of $\boldsymbol \Sigma = (\sigma_{ij})_{p\times p}\in\mathcal{U}(2,\tau,\epsilon)$ for a $10\times 10$ matrix data vectorized in the column-major order, with the entries satisfies $\sigma_{ij}=\prod_{\ell=1}^2(1+\delta_{ij\ell})^{-\alpha_\ell-1}$ for $\alpha_1=0.1$ and $\alpha_2=0.2$. The gray region is a banded area $|i-j|\le k$ for the banding estimator with a banding width $k<p_1$, and the red colored bands illustrate the covariances between adjacent grids along rows, which decay rather slowly that prevents the bandable condition in (2.3).

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• Theorem 1
• Example : Polynomially decayed covariances
• Example : Exponentially decayed covariances
• Remark
• Proposition 1
• Theorem 2
• Remark
• Theorem 3