低秩张量完备化的张量p-收缩核范数
Tensor p-shrinkage nuclear norm for low-rank tensor completion
摘要：
基于张量奇异值分解（t-SVD），提出了一种新的张量p收缩核范数（p-TNN）的定义.特别地，可以证明当p < 1时，p-TNN比张量核范数更好地逼近张量平均秩。因此，通过使用p-收缩核范数，提出了一种新的低秩张量完备（LRTC）模型，从其部分观测值估计张量。在统计上，给出了LRTC模型的恢复误差上界。此外，一个有效的算法，加速的自适应动量计划，开发解决由此产生的非凸优化问题。在光滑性假设下，可以进一步保证算法具有全局收敛速度。在合成数据集和真实数据集上进行的数值实验验证了我们的结果，并证明了我们的p-TNN在LRTC问题中的优越性。
英文摘要：
In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank than the tensor nuclear norm when p < 1. Therefore, by employing the p-shrinkage nuclear norm, a novel low-rank tensor completion (LRTC) model is proposed to estimate a tensor from its partial observations. Statistically, the upper bound of recovery error is provided for the LRTC model. Furthermore, an efficient algorithm, accelerated by the adaptive momentum scheme, is developed to solve the resulting nonconvex optimization problem. It can be further guaranteed that the algorithm enjoys a global convergence rate under the smoothness assumption. Numerical experiments conducted on both synthetic and real-world data sets verify our results and demonstrate the superiority of our p-TNN in LRTC problems over several state-of-the-art methods.
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