Gradient descent on the Stiefel manifold recovers Tucker and tensor-train tensors with linear convergence whose initialization requirement and rate scale polynomially with the tensor order N.
Tensor train completion: local recovery guarantees via Riemannian optimization
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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.
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A Scalable Factorization Approach for High-Order Structured Tensor Recovery
Gradient descent on the Stiefel manifold recovers Tucker and tensor-train tensors with linear convergence whose initialization requirement and rate scale polynomially with the tensor order N.