We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

This paper addresses classification problems with matrix-valued data, which commonly arises in applications such as neuroimaging and signal processing. Building on the assumption that the data from each class follows a matrix normal distribution, we propose a novel extension of Fisher's Linear Discriminant Analysis (LDA) tailored for matrix-valued observations. To effectively capture structural information while maintaining estimation flexibility, we adopt a nonparametric empirical Bayes framework based on Nonparametric Maximum Likelihood Estimation (NPMLE), applied to vectorized and scaled matrices. The NPMLE method has been shown to provide robust, flexible, and accurate estimates for vector-valued data with various structures in the mean vector or covariance matrix. By leveraging its strengths, our method is effectively generalized to the matrix setting, thereby improving classification performance. Through extensive simulation studies and real data applications, including electroencephalography (EEG) and magnetic resonance imaging (MRI) analysis, we demonstrate that the proposed method consistently outperforms existing approaches across a variety of data structures.

This paper addresses hypothesis testing for the mean of matrix-valued data in high-dimensional settings. We investigate the minimum discrepancy test, originally proposed by Cragg (1997), which serves as a rank test for lower-dimensional matrices. We evaluate the performance of this test as the matrix dimensions increase proportionally with the sample size, and identify its limitations when matrix dimensions significantly exceed the sample size. To address these challenges, we propose a new test statistic tailored for high-dimensional matrix rank testing. The oracle version of this statistic is analyzed to highlight its theoretical properties. Additionally, we develop a novel approach for constructing a sparse singular value decomposition (SVD) estimator for singular vectors, providing a comprehensive examination of its theoretical aspects. Using the sparse SVD estimator, we explore the properties of the sample version of our proposed statistic. The paper concludes with simulation studies and two case studies involving surveillance video data, demonstrating the practical utility of our proposed methods.

Covariance estimation for matrix-valued data has received an increasing interest in applications including neuroscience and environmental studies. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating the banded and tapering covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are studied and compared to the ones for the usual vector-valued data. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with the potential heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from an electroencephalography study and a gridded temperature anomalies dataset.