Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor $\mathcal{T}\in\mathbb{R}^{n\times n\times n}$ of low multilinear rank from partially observed entries. Riemannian optimization algorithms are a class of efficient methods for this problem, but the theoretical convergence analysis is still lacking. In this manuscript, we establish the entrywise convergence of the vanilla Riemannian gradient method for low Tucker-rank tensor completion under the nearly optimal sampling complexity $O(n^{3/2})$. Meanwhile, the implicit regularization phenomenon of …

algorithms analysis convergence low optimization paper regularization tensor tucker