Multidimensional Data Analysis Based on Block Convolutional Tensor Decomposition

TL;DR

This paper introduces a novel block convolutional tensor decomposition based on the $ ext{ extsterling}_c$-Product, offering lower complexity and improved results in high-dimensional data analysis tasks.

Contribution

It proposes a new tensor-tensor product and decomposition method based on block convolution with reflective boundary conditions, extending to arbitrary order tensors.

Findings

Lower computational complexity compared to t-SVD

Higher-quality results in classification tasks

Effective tensor analysis for high-dimensional data

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.