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The two most well-known generalizations of matrix rank to higher-order
tensors are
multilinear rank and (outer product) rank, respectively. The computation
of the best low
multilinear rank approximation of a tensor is sometimes called Tucker
compression. The two
main applications are compression of multi-way data and the computation of
dominant
subspaces of a higher-order tensor. In this talk, we touch on some new
algorithms and discuss
the problem of local minima. The (outer product) rank of a higher-order
tensor is the minimal
number of rank-1 terms in which it can be decomposed. The decomposition
itself is sometimes
called the Canonical or Parallel Factor decomposition
(CANDECOMP/PARAFAC)(CP). The most
well-known condition under which essential uniqueness of the CP
decomposition is guaranteed,
is the famous Kruskal bound. The rank is usually determined by means of
trial-and-error or by
using heuristic criteria. The most popular algorithm is of the alternating
least squares type.
However, in many practical cases of interest, the rank is equal to the
matrix rank of a matrix
representation of the tensor. In the case of exact data, CP may be
computed by means of
standard linear algebra under certain conditions on the rank. In the case
of noisy data, one can
often use algorithms for simultaneous matrix diagonalization. There also
exists a uniqueness
condition that is significantly more relaxed than Kruskal's, provided one
of the tensor dimensions
is not smaller than the tensor rank. The latter condition is satisfied in
many applications. Recently, we have
proposed Block Term Decompositions (BTD) as a unifying framework for CP
and Tucker. BTD
also unifies multilinear rank and (outer product) rank. We discuss BTD
generalizations of our CP
results.
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