The Generalized Tensor Decomposition with Heterogeneous Tensor Product for Third-Order Tensors

Recently, tensor decompositions have attracted increasing attention and shown promising performance in processing multi-dimensional data. However, the existing tensor decompositions assume that the correlation along one mode is homogeneous and thus cannot characterize the multiple types of correlations (i.e., heterogeneous correlation) along the mode in real data. To address this issue, we propose a heterogeneous tensor product that allows us to explore this heterogeneous correlation, which can degenerate into the classic tensor products (e.g., mode product and tensor–tensor product). Equipped with this heterogeneous tensor product, we develop a generalized tensor decomposition (GTD) framework for third-order tensors, which not only induces many novel tensor decompositions but also helps us to better understand the interrelationships between the new tensor decompositions and the existing tensor decompositions. Especially, under the GTD framework, we find that new tensor decompositions can faithfully characterize the multiple types of correlations along the mode. To examine the effectiveness of the new tensor decomposition, we evaluate its performance on a representative data compression task. Extensive experimental results on multispectral images, light field images, and videos compression demonstrate the superior performance of our developed tensor decomposition compared to other existing tensor decompositions.