Tensor train completion: local recovery guarantees via Riemannian optimization
This provides theoretical guarantees for tensor completion methods, which is important for applications like data recovery and machine learning, though it appears incremental as it builds on existing Riemannian optimization approaches.
The paper tackles the problem of tensor train completion by determining how many randomly selected tensor elements are needed to guarantee local convergence of Riemannian gradient descent, deriving a new bound based on singular values and introducing core coherence for tensor trains.
In this work, we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with auxiliary subspace information and obtain the corresponding local convergence guarantees.