Online Tensor Completion And Free Submodule Tracking With The T-Svd

We propose a new online algorithm, called TOUCAN, for the tensor completion problem of imputing missing entries of a low tubal-rank tensor using the 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 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 batch tensor completion algorithms and matrix completion algorithms. We show our results on real applications to recover temporal MRI data under limited sampling.