Entrywise Convergence of Riemannian Gradient Method for Low Rank Tensor Completion via Tucker Decomposition

This paper is concerned with the Tucker decomposition based low rank tensor completion problem, which is about reconstructing a tensor $mathcal{T}inmathbb{R}^{ntimes ntimes n}$ of a small multilinear rank from partially observed entries. We study the convergence of the Riemannian gradient method for this problem. Guaranteed linear convergence in terms of the infinity norm has been established for this algorithm provided the number of observed entries is essentially in the order of $O(n^{3/2})$. The convergence analysis relies on the leave-one-out technique and the subspace projection structure within the algorithm. To the best of our knowledge, this is the first work that has established the entrywise convergence of a non-convex algorithm for low rank tensor completion via Tucker decomposition.