New Theoretical Foundations of Tensor Applications: Clustering, Error Analysis, Global Convergence, and Robust Formulations

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New Theoretical Foundations of Tensor Applications: Clustering, Error Analysis, Global Convergence, and Robust FormulationsTensor decompositions become increasingly important in analyzing high-dimensional multi-index data. However, applications of tensor decompositions are so far restricted: (1) they are mainly used for data compression ? critically important tasks such as data clustering have not been addressed. (2) No bounds on reconstruction error exist ? the compression parameters are determined on a trial-and-error basis. (3) As solutions to non-convex optimizations, tensor decompositions are not unique. This could severely affect the reliability of tensor analysis. (4) Tensor decompositions are obtained via minimizing the sum of squared errors, thus are prone to noise or outliers in the data. A robust formulation of decomposition is highly desirable for applications with large noises. In this proposal, we investigate these new fundamental aspects of tensor applications: (1) Investigate the clustering capabilities of tensor decompositions, in addition to the established theoretical results on clustering; (2) Provide comprehensive error analysis of tensor decompositions and derive lower and upper error bounds; (3) Investigate conditions for global convergence for tensor decompositions and investigate good initializations for the cases where global convergence fails. (4) Develop robust formulations for tensor decompositions.In addition, we will develop user-friendly software toolbox that contains the resulting algorithms and make it available to the public. We will also educate graduate and undergraduate students with fundamentals in matrix and tensor computations. We will present tutorials and organize workshops on this new direction.