Tensors in Statistics

This article provides an overview of tensors, their properties, and their applications in statistics. Tensors, also known as multidimensional arrays, are generalizations of matrices to higher orders and are useful data representation architectures. We first review basic tensor concepts and decompositions, and then we elaborate traditional and recent applications of tensors in the fields of recommender systems and imaging analysis. We also illustrate tensors for network data and explore the relations among interacting units in a complex network system. Some canonical tensor computational algorithms and available software libraries are provided for various tensor decompositions. Future research directions, including tensors in deep learning, are also discussed.