标题： 鲁棒低秩张量回归 via 剪枝和Huber损失 显示英文标题

标题： Robust low-rank tensor regression via clipping and Huber loss

摘要： 本文中，我们基于截断方法和Huber损失构建了一个在强健低秩张量回归下的参数估计框架，并研究了在仅有有限二阶矩的随机噪声下强健低秩张量回归模型。通过梯度下降法，我们提出的Huber型强健估计器在两个方面理论上是最优的：(1) 我们的统计误差率几乎与子高斯误差下传统最小二乘法得出的最佳上界相同；(2) 恢复张量参数的样本复杂度也是最优的。大量的数值实验显示了我们估计器的鲁棒性，截断和Huber损失的使用有助于提高所提出算法的稳定性和统计有效性，这比最小二乘法更为优越。同时，通过模拟确认了所提出的强健估计器收敛速率中的相变现象。此外，我们将这种估计技术应用于图像压缩，证明了我们的方法更有效。 显示英文摘要

摘要： In this paper, we construct a parameter estimation framework under robust low-rank tensor regression based on the truncation method and Huber loss, and study robust low-rank tensor regression model under random noise with only finite second-order moment. Through the gradient descent method, our proposed Huber-type robust estimator is theoretically optimal in two aspects: (1) our statistical error rate is nearly the same as the optimal upper bound deduced by the traditional least squares method under sub-Gaussian error; (2) the sample complexity of recovering the tensor parameter is also optimal. Extensive numerical experiments show the robustness of our estimator, and the utilization of truncation and Huber loss is beneficial to improve the stability and statistical effectiveness of the proposed algorithm which is superior to the least squares method. Meanwhile, the phenomenon of phase transition in the convergence rate of the proposed robust estimator is confirmed through simulation. Furthermore, we apply this estimation technique to image compression, which demonstrates that our method is more effective.