Tensor-CUR Decompositions for Tensor-Based Data

Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensor-based extension of the matrix CUR decomposition. The tensor-CUR decomposition is most relevant as a data analysis tool when the data consist of one mode that is qualitatively different from the others. In this case, the tensor-CUR decomposition approximately expresses the original data tensor in terms of a basis consisting of underlying subtensors that are actual data elements and thus that have a natural interpretation in terms of the processes generating the data. Assume the data may be modeled as a $(2+1)$-tensor, i.e., an $m \times n \times p$ tensor $\mathcal{A}$ in which the first two modes are similar and the third is qualitatively different. We refer to each of the 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.