Hybrid CUR-type decomposition of tensors in the Tucker format

TL;DR

This paper proposes a hybrid CUR-type tensor decomposition in Tucker format that improves approximation accuracy by selectively preserving fibers across multiple modes, especially as tensor dimensions grow.

Contribution

It introduces a novel hybrid algorithm for tensor CUR decomposition that reduces approximation error compared to standard methods, with flexible fiber preservation.

Findings

Smaller approximation error than standard tensor CUR methods.

Error increases with tensor dimension and fewer preserved modes.

Flexible fiber preservation enhances approximation quality.

Abstract

The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor $\mathcal{X}$ as a product of a core tensor $\mathcal{S}$, a matrix $C$ obtained by extracting mode-$k$ fibers of $\mathcal{X}$, and matrices $U_j$, $j=1,\ldots,k-1,k+1,\ldots,d$, chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.