High Dimensional Covariance Estimation for Spatio-Temporal Processes 

Abstract
High dimensional time series and array-valued data are ubiquitous in signal processing, machine learning, and science. The total dimensionality of the data is often extremely high, requiring large numbers of training examples to learn the distribution using unstructured techniques. However, due to difficulties in sampling, small population sizes, and/or rapid system changes in time, it is often the case that very few relevant training samples are available, necessitating the imposition of structure on the data if learning is to be done.

In this work, we develop various forms of multidimensional covariance
structure that explicitly exploit the array structure of the data, in a way
analogous to the widely used low rank modeling of the mean. This allows
dramatic reductions in the number of training samples required, in some
cases to a single training sample. Using this structure, we provide strong nonasymptotic statistical performance guarantees for high dimensional estimation in sample-starved scenarios.

Contributions are made in the following areas: development of a variety of rich Kronecker product-based covariance models, strong performance bounds for high-dimensional estimation of covariances under each model, and an adaptive online estimation method for low-rank covariance metrics.