论文标题
（几乎）正交分解张量的扰动范围
Perturbation Bounds for (Nearly) Orthogonally Decomposable Tensors
论文作者
论文摘要
我们以一种类似于Weyl，Davis，Kahan和Wedin引起的矩阵的经典结果的精神，开发了正交分解张量的奇异值和向量的确定性扰动界限。我们的边界表明矩阵和高阶张量之间有着有趣的差异。最值得注意的是，它们表明，对于高阶张量，扰动会影响每个基本值/向量，并且其对基本奇异矢量的影响不取决于其相应的奇异值或与其他奇异值的距离的多重性。我们的结果很容易应用，并为涉及高阶正交分解张量的光谱学习的许多不同问题和机器学习中的许多不同问题提供了统一的处理。特别是，我们说明了界限在高维张量SVD问题的背景下的含义，以及如何使用它来得出光谱学习的最佳收敛速率。
We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds demonstrate intriguing differences between matrices and higher-order tensors. Most notably, they indicate that for higher-order tensors perturbation affects each essential singular value/vector in isolation, and its effect on an essential singular vector does not depend on the multiplicity of its corresponding singular value or its distance from other singular values. Our results can be readily applied and provide a unified treatment to many different problems in statistics and machine learning involving spectral learning of higher-order orthogonally decomposable tensors. In particular, we illustrate the implications of our bounds in the context of high dimensional tensor SVD problem, and how it can be used to derive optimal rates of convergence for spectral learning.
