Multidimensional Data Analysis Based on Block Convolutional Tensor Decomposition

• URL: http://arxiv.org/abs/2308.01768v2
• Date: Fri, 11 Aug 2023 21:23:56 GMT
• Title: Multidimensional Data Analysis Based on Block Convolutional Tensor Decomposition
• Authors: Mahdi Molavi, Mansoor Rezghi, and Tayyebeh Saeedi
• Abstract summary: We propose a new tensor-tensor product called the $star_ctext-Product$ based on Block convolution with reflective boundary conditions. We also introduce a tensor decomposition based on our $star_ctext-Product$ for arbitrary order tensors. Compared to t-SVD, our new decomposition has lower complexity, and experiments show that it yields higher-quality results in applications such as classification and compression.
• Score: 1.1674893622721483
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Tensor decompositions are powerful tools for analyzing multi-dimensional data in their original format. Besides tensor decompositions like Tucker and CP, Tensor SVD (t-SVD) which is based on the t-product of tensors is another extension of SVD to tensors that recently developed and has found numerous applications in analyzing high dimensional data. This paper offers a new insight into the t-Product and shows that this product is a block convolution of two tensors with periodic boundary conditions. Based on this viewpoint, we propose a new tensor-tensor product called the $\star_c{}\text{-Product}$ based on Block convolution with reflective boundary conditions. Using a tensor framework, 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, our new decomposition has lower complexity, and experiments show that it yields higher-quality results in applications such as classification and compression.

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