# Tensor-CUR Decompositions for Tensor-Based Data

## Abstract

*p*different $m \times n$ matrices as “slabs” and each of the $mn$ different

*p*-vectors as “fibers.” In this case, the tensor-CUR algorithm computes an approximation to the data tensor $\mathcal{A}$ that is of the form $\mathcal{CUR}$, where $\mathcal{C}$ is an $m \times n \times c$ tensor consisting of a small number

*c*of the slabs, $\mathcal{R}$ is an $r \times p$ matrix consisting of a small number

*r*of the fibers, and $\mathcal{U}$ is an appropriately defined and easily computed $c \times r$ encoding matrix. Both $\mathcal{C}$ and $\mathcal{R}$ may be chosen by randomly sampling either slabs or fibers according to a judiciously chosen and data-dependent probability distribution, and both

*c*and

*r*depend on a rank parameter

*k*, an error parameter $\epsilon$, and a failure probability $\delta$. Under appropriate assumptions, provable bounds on the Frobenius norm of the error tensor $\mathcal{A} - \mathcal{CUR}$ are obtained. In order to demonstrate the general applicability of this tensor decomposition, we apply it to problems in two diverse domains of data analysis: hyperspectral medical image analysis and consumer recommendation system analysis. In the hyperspectral data application, the tensor-CUR decomposition is used to

*compress*the data, and we show that classification quality is not substantially reduced even after substantial data compression. In the recommendation system application, the tensor-CUR decomposition is used to

*reconstruct*missing entries in a user-product-product preference tensor, and we show that high quality recommendations can be made on the basis of a small number of basis users and a small number of product-product comparisons from a new user.

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**Submitted**: 17 July 2006

**Accepted**: 8 January 2007

**Published online**: 25 September 2008

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