BIGDATA: F: DKA: CSD: DKM: Theory and Algorithms for Processing Data with Sparse and Multilinear Structure

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Tensors, or multidimensional array structures, naturally arise in big data applications such as psychometrics, multimedia, social media, genomics, neuroimaging, geospatial data, and turbulent flow simulations. Tensors in these applications have massive amounts of data and are difficult to store, transmit, compute with, and analyze. To perform these tasks successfully it is crucial to discover and utilize relevant and informative structure within the data, which, in the tensor context, often has the form of a decomposition or factorization into a relatively small number of simpler constituents. Unfortunately, compared to the simpler matrix case, existing mathematical theory and computational tools for tensors are severely limited, and do not meet the needs of big data applications. The broad goal of this project is to close this theoretical and practical gap in tools for tensors for big data. Because of the fundamental and ubiquitous role of tensors in Big Data, the results of this research have the potential to impact every field in which big data is of interest. In particular, the new tools and methodology have the potential to enable new fundamental discoveries in neuroscience, with important benefits to human health.More specifically, this research aims for computationally efficient algorithms with theoretical performance guarantees. The work emphasizes highly scalable online and distributed versions, approaching the fundamental limits. Approaches include relaxations to achieve low rank decomposition, identifying fundamental limits for compressed sensing, and consideration of practical issues such as noisy data. These algorithms are validated on big data applications in multimodality functional neuroimaging and neuroscience. As it draws on and includes mathematics, computer science, engineering, statistics, and neuroscience, this research is highly multidisciplinary.