SummaryThe 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.