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  1. We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a lowtubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling.
  2. We propose a new online algorithm, called TOUCAN, forthe tensor completion problem of imputing missing entriesof a low tubal-rank tensor using the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) al-gebraic framework. We also demonstrate TOUCAN’s abilityto track changing free submodules from highly incompletestreaming 2-D data. TOUCAN uses principles from incre-mental gradient descent on the Grassmann manifold to solvethe tensor completion problem with linear complexity andconstant memory in the number of time samples. We com-pare our results to state-of-the-art batch tensor completion al-gorithms and matrix completion algorithms. We show our re-sults on real applications to recover temporal MRI data underlimited sampling.