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

Understand

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor $ 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 the algorithm has also been revealed.