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$^6$.

翻译：本文研究高压高压SVD的一般框架。 我们提出一种新的计算效率算法,即高压列电或高振动迭代(TTOI),目的是估计高压高压观测产生的低压压列级结构; 拟议的TTOI包括通过TT-SVD(Oseledets,2011年)和新的迭代后向/前向更新启动; 我们开发了TTOI估算误差的总上限,在数个关于拉子进制的新代表列母体的支持下,我们开发了一个匹配的信息理论下限(TTOI),我们还证明TTOI在高压拉子模型下实现了微模最佳性; 拟议的TTOI的优点是通过对高压马科夫进程进行估计和尺寸削减的应用、数字研究以及纽约市出租车旅行记录的一个真实数据实例来说明的。 提议的算法软件可以在线获得6美元。