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Tensor decompositions are powerful tools for analyzing multi-dimensional datain their original format. Besides tensor decompositions like Tucker and CP,Tensor SVD (t-SVD) which is based on the t-product of tensors is anotherextension of SVD to tensors that recently developed and has found numerousapplications in analyzing high dimensional data. This paper offers a newinsight into the t-Product and shows that this product is a block convolutionof two tensors with periodic boundary conditions. Based on this viewpoint, wepropose a new tensor-tensor product called the $\star_c{}\text{-Product}$ basedon Block convolution with reflective boundary conditions. Using a tensorframework, this product can be easily extended to tensors of arbitrary order.Additionally, we introduce a tensor decomposition based on our$\star_c{}\text{-Product}$ for arbitrary order tensors. Compared to t-SVD, ournew decomposition has lower complexity, and experiments show that it yieldshigher-quality results in applications such as classification and compression.

Multidimensional Data Analysis Based on Block Convolutional Tensor Decomposition