论文标题
通过张量训练正交迭代的最佳高级张量SVD
Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration
论文作者
论文摘要
本文研究了高阶张量SVD的一般框架。我们提出了一种新的计算有效算法，张量 - 训练正交迭代（TTOI），旨在从嘈杂的高阶张量观察中估算低张量训练秩的结构。拟议的TTOI包括通过TT-SVD初始化（Oseledets，2011）和新的迭代向后/正更新。我们在张张量进程中的几个新表示引理的支持下，在TTOI的估计误差上开发了一般的上限。通过开发匹配的信息理论下限，我们还证明TTOI在尖峰张量模型下实现了最小值的最佳性。拟议的TTOI的优点通过应用于估计和尺寸缩小高阶马尔可夫流程，数值研究以及纽约市出租车旅行记录的真实数据示例。该算法的软件可在线获得$^6 $。
This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD (Oseledets, 2011) and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online$^6$.
