  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.
