In this paper a new Riemannian rank adaptive method (RRAM) is proposed for the low-rank tensor completion problem (LRTCP) formulated as a least-squares optimization problem on the algebraic variety of tensors of bounded tensor-train (TT) rank. The method iteratively optimizes over fixed-rank smooth manifolds using a Riemannian conjugate gradient algorithm from Steinlechner (2016) and gradually increases the rank by computing a descent direction in the tangent cone to the variety. Additionally, a numerical method to estimate the amount of rank increase is proposed based on a theoretical result for the stationary points of the low-rank tensor approximation problem and a definition of an estimated TT-rank. Furthermore, when the iterate comes close to a lower-rank set, the RRAM decreases the rank based on the TT-rounding algorithm from Oseledets (2011) and a definition of a numerical rank. We prove that the TT-rounding algorithm can be considered as an approximate projection onto the lower-rank set which satisfies a certain angle condition to ensure that the image is sufficiently close to that of an exact projection. Several numerical experiments are given to illustrate the use of the RRAM and its subroutines in {\Matlab}. Furthermore, in all experiments the proposed RRAM outperforms the state-of-the-art RRAM for tensor completion in the TT format from Steinlechner (2016) in terms of computation time.

翻译：本文针对有界张量列车（TT）秩的张量代数簇上的低秩张量补全问题（LRTCP），提出了一种新的黎曼秩自适应方法（RRAM）。该方法采用Steinlechner（2016）的黎曼共轭梯度算法，在固定秩光滑流形上迭代优化，并通过计算切锥内的下降方向逐步增加秩。此外，基于低秩张量近似问题驻点的理论结果以及估计TT秩的定义，提出了一种估计秩增量大小的数值方法。当迭代点接近低秩集合时，RRAM利用Oseledets（2011）的TT舍入算法和数值秩的定义降低秩。我们证明TT舍入算法可视为满足特定角度条件的低秩集合近似投影，从而确保投影后的像与精确投影充分接近。通过多个数值实验展示了RRAM及其子程序在{\Matlab}中的应用。所有实验结果表明，与Steinlechner（2016）提出的TT格式张量补全的当前最优RRAM相比，本文方法在计算时间上更具优势。