低Tucker秩张量补全的黎曼优化隐式正则化与逐元素收敛性
Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion
AI总结：
本文研究了低Tucker秩张量补全问题，在近乎最优的采样复杂度下，利用留一法技术证明了原始黎曼梯度方法的逐元素收敛性并揭示了其隐式正则化现象。
摘要：
本文研究低Tucker秩张量补全问题，即从部分观测到的元素重构一个具有低多线性秩的张量 $ T \in\mathbb{R}^{n\times n \times n}$。黎曼优化算法是解决该问题的一类高效方法，但理论收敛性分析仍然缺乏。在本文中，我们在近乎最优的采样复杂度 $O(n^{3/2})$ 下，建立了用于低Tucker秩张量补全的原始黎曼梯度方法的逐元素收敛性。同时，还揭示了该算法的隐式正则化现象。据我们所知，这是首个针对低Tucker秩张量补全的非凸方法，证明其具有逐元素收敛性和隐式正则化性质的工作。该分析依赖于留一法技术，本文得出的一些技术结果在研究该问题其他非凸方法的性质时可能具有更广泛的意义。
英文摘要：
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. As far as we know, this is the first work that has shown the entrywise convergence and implicit regularization property of a non-convex method for low Tucker-rank tensor completion. The analysis relies on the leave-one-out technique, and some of the technical results developed in the paper might be of broader interest in investigating the properties of other non-convex methods for this problem.
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