Grassmannian Optimization for Online Tensor Completion and Tracking in the t-SVD Algebra

We propose a new streaming algorithm, called TOUCAN, for the tensor completion problem of imputing missing entries of a low-tubal-rank tensor using the recently proposed tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) algebraic framework. We also demonstrate TOUCAN's ability to track changing free submodules from highly incomplete streaming 2-D data. TOUCAN 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 compare our results to state-of-the-art tensor completion algorithms in real applications to recover temporal chemo-sensing data and MRI data under limited sampling.