Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD
This work addresses efficient online tensor completion for real-time data processing in domains like medical imaging and sensing, though it is incremental as it builds on existing t-SVD and Grassmannian optimization methods.
The authors tackled the problem of imputing missing entries in low-tubal-rank tensors from streaming data by proposing a fast algorithm using the t-SVD framework, which achieved competitive accuracy and significantly faster compute times than state-of-the-art methods in applications like chemo-sensing and MRI data recovery.
We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a low-tubal-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.