The higher order singular value decomposition, which is regarded as a generalization of the matrix singular value decomposition (SVD), has a long history and is well established, while the T‐SVD is relatively new and lacks systematic analysis. Because of the unusual tensor‐tensor product that the T‐SVD is based on, the form of the T‐SVD may be difficult to comprehend. The main aim of this article is to establish a connection between these two decompositions. By converting the form of the T‐SVD into the sum of outer product terms, we compare the forms of the two decompositions. Moreover, from establishing the connection, a new decomposition which has a specific nonzero pattern, is proposed and developed. Numerical examples are given to demonstrate the useful ability of the new decomposition for approximation and data compression.

中文翻译：

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三阶张量的分解：HOSVD，T‐SVD和Beyond

高阶奇异值分解被认为是矩阵奇异值分解（SVD）的推广，它历史悠久且建立良好，而T‐SVD相对较新并且缺乏系统分析。由于T‐SVD基于不寻常的张量-张量积，因此T‐SVD的形式可能难以理解。本文的主要目的是在这两个分解之间建立联系。通过将T-SVD的形式转换为外部乘积项的总和，我们比较了两种分解的形式。此外，通过建立连接，提出并开发了具有特定非零模式的新分解。数值例子说明了新分解对于逼近和数据压缩的有用能力。