Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. \emph{Tensor power method} (TPM) is one of the widely-used techniques in the decomposition of tensors. This paper presents a novel tensor power method for decomposing arbitrary order tensors, which overcomes limitations of existing approaches that are often restricted to lower-order (less than $3$) tensors or require strong assumptions about the underlying data structure. We apply sketching method, and we are able to achieve the running time of $\widetilde{O}(n^{p-1})$, on the power $p$ and dimension $n$ tensor. We provide a detailed analysis for any $p$-th order tensor, which is never given in previous works.

翻译：张量分解是处理高维数据的各个领域中的一种基本方法。*张量幂方法*（TPM）是张量分解中广泛使用的技术之一。本文提出了一种新的张量幂方法，用于分解任意阶张量，克服了现有方法通常局限于低阶（小于$3$）张量或需要对底层数据结构做出强假设的局限性。我们应用了草图化技术，并实现了对幂次为$p$、维度为$n$的张量在$p$和$n$上的运行时间达到$\widetilde{O}(n^{p-1})$。针对任意$p$阶张量，我们提供了此前工作中从未给出的详细分析。