Tensors in High-Dimensional Data Analysis: Methodological Opportunities and Theoretical Challenges

Large amounts of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization, and numerical linear algebra, among other fields. Despite these hurdles, significant progress has been made in the past decade. This review seeks to examine some of the key advancements and identify common threads among them, under a number of different statistical settings.