The Sup-norm Perturbation of HOSVD and Low Rank Tensor Denoising
The Sup-norm Perturbation of HOSVD and Low Rank Tensor Denoising
复制标题
HOSVD的超范数扰动与低阶张量去噪
DOI：
--
复制
发表时间：
2017
影响因子：
6
通讯作者：
Fan Zhou
中科院分区：
文献类型：
--
作者：
Dong Xia;Fan Zhou
张量的高阶奇异值分解（HOSVD）是矩阵SVD的推广。HOSVD在随机噪声下的扰动分析比矩阵方法更精细。最近，多项式时间算法已被提出，其中奇异子空间和低秩张量的统计最优估计是可达到的欧氏范数。在本文中，我们分析了HOSVD的上范数扰动界，并引入了在上范数下具有锐偏差界的奇异子空间的估计。我们还研究了一个低秩张量去噪估计，并证明了其快速收敛速度的条目方面的错误。超范数扰动界揭示了统计学习应用中的非常规相变，例如高维高斯混合模型中的精确聚类和子张量局部化中的精确支持度恢复。此外，HOSVD的上界也给出了非平衡（或胖）矩阵奇异子空间的单侧超范数扰动上界。
The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces and the low rank tensors are attainable in the Euclidean norm. In this article, we analyze the sup-norm perturbation bounds of HOSVD and introduce estimators of the singular subspaces with sharp deviation bounds in the sup-norm. We also investigate a low rank tensor denoising estimator and demonstrate its fast convergence rate with respect to the entry-wise errors. The sup-norm perturbation bounds reveal unconventional phase transitions for statistical learning applications such as the exact clustering in high dimensional Gaussian mixture model and the exact support recovery in sub-tensor localizations. In addition, the bounds established for HOSVD also elaborate the one-sided sup-norm perturbation bounds for the singular subspaces of unbalanced (or fat) matrices.
影响因子：
7
作者：
Xiong, Qing;Ancona, Nicola;Furey, Terrence S.
通讯作者：
Furey, Terrence S.