  In many applications such as data compression, imaging or genomic data
analysis, it is important to approximate a given tensor by a tensor that is
sparsely representable. For matrices, i.e. 2-tensors, such a representation can
be obtained via the singular value decomposition, which allows to compute best
rank k-approximations. For very big matrices a low rank approximation using SVD
is not computationally feasible. In this case different approximations are
available. It seems that variants of CUR decomposition are most suitable. For
d-mode tensors T with d>2, many generalizations of the singular value
decomposition have been proposed to obtain low tensor rank decompositions. The
most appropriate approximation seems to be best (r_1,...,r_d)-approximation,
which maximizes the l_2 norm of the projection of T on a tensor product of
subspaces U_1,...,U_d, where U_i is an r_i-dimensional subspace. One of the
most common method is the alternating maximization method (AMM). It is obtained
by maximizing on one subspace U_i, while keeping all other fixed, and
alternating the procedure repeatedly for i=1,...,d. Usually, AMM will converge
to a local best approximation. This approximation is a fixed point of a
corresponding map on Grassmannians. We suggest a Newton method for finding the
corresponding fixed point. We also discuss variants of CUR-approximation method
for tensors. The first part of the paper is a survey on low rank approximation
of tensors. The second new part of this paper is a new Newton method for best
$(r_1,...,r_d)$-approximation. We compare numerically different approximation
methods.
