Large amounts of multidimensional data in the form of multilinear arrays, or tensors, arise routinely in modern applications from such diverse fields as chemometrics, genomics, physics, psychology, and signal processing, among many others. At the present time, our ability to generate and acquire such data has far outpaced our ability to effectively extract useful information. There is a clear need to develop novel statistical methods, efficient computational algorithms, and fundamental mathematical theory to analyze and exploit information in these types of data. This research project builds upon prior work in high-dimensional statistics, genomics, quantum physics, fast functional MRI, and closed-loop diabetes control to address the challenges in analysis of large tensorial data sets. It is anticipated that the project will help to advance future research in these and other areas of applications.
One of the main challenges in dealing with this type of data is to develop methods of statistical inference to achieve both the statistical and computational efficiencies. More often than not, these two aims are not simultaneously achieved by existing methods: there is a gap between the optimal rate in the minimax sense and the best rate achievable by existing polynomial-time algorithms. The overarching goal of this project is to develop statistical methods, algorithms, and theory to efficiently, both statistically and computationally, analyze large scale data in the form of tensors. In particular, four most common and interrelated problems will be studied systematically: low-rank tensor denoising, low-rank tensor regression, estimation of the moment tensor, and tensor phase retrieval.