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.

翻译：本文研究从大量含噪线性测量中估计低Tucker秩张量的问题。该一般性问题涵盖应用中出现的许多具体实例，包括张量回归、张量补全以及张量主成分分析/奇异值分解。我们针对低Tucker秩张量估计提出一种高效的黎曼高斯-牛顿方法。与文献中此类方法常规的（超）线性收敛性保证不同，我们首次证明了在含噪条件下，特定正则性假设下RGN方法具有局部二次收敛性，并给出了相应的估计误差上界。通过建立与上界相匹配的确定性估计误差下界，证实了RGN方法的统计最优性。我们通过两个机器学习应用（张量回归与张量SVD）展示了该方法的优势，最后通过仿真结果验证了理论发现。