We consider a Canonical Polyadic (CP) decomposition approach to low-rank tensor completion (LRTC) by incorporating external pairwise similarity relations through graph Laplacian regularization on the CP factor matrices. The usage of graph regularization entails benefits in the learning accuracy of LRTC, but at the same time, induces coupling graph Laplacian terms that hinder the optimization of the tensor completion model. In order to solve graph-regularized LRTC, we propose efficient alternating minimization algorithms by leveraging the block structure of the underlying CP decomposition-based model. For the subproblems of alternating minimization, a linear conjugate gradient subroutine is specifically adapted to graph-regularized LRTC. Alternatively, we circumvent the complicating coupling effects of graph Laplacian terms by using an alternating directions method of multipliers. Based on the Kurdyka-{\L}ojasiewicz property, we show that the sequence generated by the proposed algorithms globally converges to a critical point of the objective function. Moreover, the complexity and convergence rate are also derived. In addition, numerical experiments including synthetic data and real data show that the graph regularized tensor completion model has improved recovery results compared to those without graph regularization, and that the proposed algorithms achieve gains in time efficiency over existing algorithms.

翻译：我们考虑通过图拉普拉斯正则化在CP因子矩阵上引入外部成对相似关系，实现低秩张量完备化（LRTC）的规范多元分解（CP）方法。图正则化的使用有助于提升LRTC的学习精度，但同时也引入了耦合的图拉普拉斯项，增加了张量完备化模型优化的难度。为求解图正则化LRTC问题，我们提出基于CP分解模型的块结构的高效交替极小化算法。针对交替极小化的子问题，我们专门设计了适用于图正则化LRTC的线性共轭梯度子程序。此外，我们采用交替方向乘子法回避图拉普拉斯项的复杂耦合效应。基于Kurdyka-Łojasiewicz性质，我们证明了所提算法生成的序列全局收敛至目标函数的临界点，并导出了算法的复杂度与收敛率。数值实验（包含合成数据与真实数据）表明：相较于无图正则化的方法，图正则化张量完备化模型具有更优的恢复效果，且所提算法相比现有算法在时间效率上更具优势。