Provable Tensor-Train Format Tensor Completion by Riemannian Optimization

• URL: http://arxiv.org/abs/2108.12163v1
• Date: Fri, 27 Aug 2021 08:13:58 GMT
• Title: Provable Tensor-Train Format Tensor Completion by Riemannian Optimization
• Authors: Jian-Feng Cai, Jingyang Li, Dong Xia
• Abstract summary: We provide the first theoretical guarantees of the convergence of RGrad algorithm for TT-format tensor completion. We also propose a novel approach, referred to as the sequential second-order moment method.
• Score: 22.166436026482984
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The tensor train (TT) format enjoys appealing advantages in handling structural high-order tensors. The recent decade has witnessed the wide applications of TT-format tensors from diverse disciplines, among which tensor completion has drawn considerable attention. Numerous fast algorithms, including the Riemannian gradient descent (RGrad) algorithm, have been proposed for the TT-format tensor completion. However, the theoretical guarantees of these algorithms are largely missing or sub-optimal, partly due to the complicated and recursive algebraic operations in TT-format decomposition. Moreover, existing results established for the tensors of other formats, for example, Tucker and CP, are inapplicable because the algorithms treating TT-format tensors are substantially different and more involved. In this paper, we provide, to our best knowledge, the first theoretical guarantees of the convergence of RGrad algorithm for TT-format tensor completion, under a nearly optimal sample size condition. The RGrad algorithm converges linearly with a constant contraction rate that is free of tensor condition number without the necessity of re-conditioning. We also propose a novel approach, referred to as the sequential second-order moment method, to attain a warm initialization under a similar sample size requirement. As a byproduct, our result even significantly refines the prior investigation of RGrad algorithm for matrix completion. Numerical experiments confirm our theoretical discovery and showcase the computational speedup gained by the TT-format decomposition.

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