Supervised classifiers of ultra high-dimensional higher-order data with locally doubly exchangeable covariance structure

We explore the performance accuracy of the linear and quadratic classifiers for ultra highdimensional higher-order data, assuming that the class conditional distributions are multivariate normal with locally doubly exchangeable covariance structure. We derive a two-stage procedure for estimating the covariance matrix: at the first stage, the Lasso-based structure learning is applied to sparsifying the block components within the covariance matrix. At the second stage, the maximum likelihood estimators of all block-wise parameters are derived given that the within block covariance structure is doubly exchangeable and the mean vector has a Kronecker product structure. We also study the effect of the block size on the classification performance in the ultra high-dimensional setting and derive a class of asymptotically equivalent block structure approximations, in a sense that the choice of the block size is asymptotically negligible. Using synthetic data, we have shown that our new supervised decision rules are very efficient in learning by very small sized training samples and then successfully classifying the test samples.