Hybrid CUR-type decomposition of tensors in the Tucker format

TL;DR

The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format that can easily be modified to preserve the fibers in more than one mode.

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.

Figures (4)

• Figure 1: Relative approximation error when tensor order varies.
• Figure 2: Relative approximation error when the number of modes in which the fibers of the original tensors are preserved varies.
• Figure 3: Relative approximation error for random tensors when the approximation rank varies.
• Figure 4: Relative approximation error for a random matrix when the approximation rank varies.

Theorems & Definitions (1)

• proof