Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence
Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence
复制标题
DOI：
--
复制
发表时间：
2021-04
期刊：
影响因子：
--
通讯作者：
Yuetian Luo;Anru R. Zhang
中科院分区：
文献类型：
--
作者：
Yuetian Luo;Anru R. Zhang
在本文中，我们考虑了从一些有噪声的线性测量中估计一个低塔克秩张量。一般问题涵盖了应用中产生的许多具体示例，包括张量回归、张量补全和张量PCA/SVD。我们考虑了一种有效的黎曼高斯-牛顿（RGN）方法用于低塔克秩张量估计。与文献中RGN的一般（超）线性收敛保证不同，我们在一定的正则性条件下证明了RGN在噪声环境下低秩张量估计的第一个局部二次收敛保证，并给出了相应的估计误差上界。给出了一个与上界匹配的确定性估计误差下界，证明了RGN的统计最优性。RGN的优点通过两个机器学习应用来说明：张量回归和张量SVD。最后，我们提供了仿真结果来证实我们的理论发现。
In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We consider an efficient Riemannian Gauss-Newton (RGN) method for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first local quadratic convergence guarantee of RGN for low-rank tensor estimation in the noisy setting under some regularity conditions and provide the corresponding estimation error upper bounds. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.