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Robust Low-Rank Tensor Recovery With Regularized Redescending M-Estimator

机译：稳健的低阶张量恢复和正则化的M估计器

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摘要

This paper addresses the robust low-rank tensor recovery problems. Tensor recovery aims at reconstructing a low-rank tensor from some linear measurements, which finds applications in image processing, pattern recognition, multitask learning, and so on. In real-world applications, data might be contaminated by sparse gross errors. However, the existing approaches may not be very robust to outliers. To resolve this problem, this paper proposes approaches based on the regularized redescending M-estimators, which have been introduced in robust statistics. The robustness of the proposed approaches is achieved by the regularized redescending M-estimators. However, the nonconvexity also leads to a computational difficulty. To handle this problem, we develop algorithms based on proximal and linearized block coordinate descent methods. By explicitly deriving the Lipschitz constant of the gradient of the data-fitting risk, the descent property of the algorithms is present. Moreover, we verify that the objective functions of the proposed approaches satisfy the Kurdyka-Łojasiewicz property, which establishes the global convergence of the algorithms. The numerical experiments on synthetic data as well as real data verify that our approaches are robust in the presence of outliers and still effective in the absence of outliers.
机译：本文解决了鲁棒的低秩张量恢复问题。张量恢复的目的是从一些线性测量中重建低阶张量，从而在图像处理，模式识别，多任务学习等方面找到应用。在实际应用中，稀疏的严重错误可能会污染数据。但是，现有方法对于异常值可能不是很健壮。为了解决此问题，本文提出了基于正则化降序M估计量的方法，这些方法已在稳健统计中引入。所提出方法的鲁棒性是通过正则化降序M估计器实现的。但是，非凸性也导致计算困难。为了解决这个问题，我们开发了基于近端和线性化块坐标下降方法的算法。通过显式推导数据拟合风险梯度的Lipschitz常数，可以得出算法的下降特性。此外，我们验证了所提出方法的目标函数满足Kurdyka-Łojasiewicz性质，从而建立了算法的全局收敛性。对合成数据和真实数据进行的数值实验证明，我们的方法在存在异常值时是鲁棒的，而在没有异常值时仍然有效。

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