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Rank minimization on tensor ring: an efficient approach for tensor decomposition and completion
Machine Learning ( IF 7.5 ) Pub Date : 2019-11-04 , DOI: 10.1007/s10994-019-05846-7
Longhao Yuan , Chao Li , Jianting Cao , Qibin Zhao

In recent studies, tensor ring decomposition (TRD) has become a promising model for tensor completion. However, TRD suffers from the rank selection problem due to the undetermined multilinear rank. For tensor decomposition with missing entries, the sub-optimal rank selection of traditional methods leads to the overfitting/underfitting problem. In this paper, we first explore the latent space of the TRD and theoretically prove the relationship between the TR-rank and the rank of the tensor unfoldings. Then, we propose two tensor completion models by imposing the different low-rank regularizations on the TR-factors, by which the TR-rank of the underlying tensor is minimized and the low-rank structures of the underlying tensor are exploited. By employing the alternating direction method of multipliers scheme, our algorithms obtain the TR factors and the underlying tensor simultaneously. In experiments of tensor completion tasks, our algorithms show robustness to rank selection and high computation efficiency, in comparison to traditional low-rank approximation algorithms.

中文翻译:

张量环上的秩最小化:一种有效的张量分解和完成方法

在最近的研究中,张量环分解(TRD)已成为张量完成的有前途的模型。然而,由于不确定的多线性秩,TRD 会遇到秩选择问题。对于缺少条目的张量分解,传统方法的次优秩选择导致过拟合/欠拟合问题。在本文中,我们首先探索了 TRD 的潜在空间,并从理论上证明了 TR-rank 与张量展开的 rank 之间的关系。然后,我们通过对 TR 因子施加不同的低秩正则化来提出两种张量完成模型,通过这些模型最小化底层张量的 TR 秩并利用底层张量的低秩结构。通过采用乘法器方案的交替方向法,我们的算法同时获得 TR 因子和基础张量。在张量完成任务的实验中,与传统的低秩逼近算法相比,我们的算法表现出对秩选择的鲁棒性和高计算效率。
更新日期:2019-11-04
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