11,964 research outputs found
Low-rank approximate inverse for preconditioning tensor-structured linear systems
In this paper, we propose an algorithm for the construction of low-rank
approximations of the inverse of an operator given in low-rank tensor format.
The construction relies on an updated greedy algorithm for the minimization of
a suitable distance to the inverse operator. It provides a sequence of
approximations that are defined as the projections of the inverse operator in
an increasing sequence of linear subspaces of operators. These subspaces are
obtained by the tensorization of bases of operators that are constructed from
successive rank-one corrections. In order to handle high-order tensors,
approximate projections are computed in low-rank Hierarchical Tucker subsets of
the successive subspaces of operators. Some desired properties such as symmetry
or sparsity can be imposed on the approximate inverse operator during the
correction step, where an optimal rank-one correction is searched as the tensor
product of operators with the desired properties. Numerical examples illustrate
the ability of this algorithm to provide efficient preconditioners for linear
systems in tensor format that improve the convergence of iterative solvers and
also the quality of the resulting low-rank approximations of the solution
Uniqueness of Nonnegative Tensor Approximations
We show that for a nonnegative tensor, a best nonnegative rank-r
approximation is almost always unique, its best rank-one approximation may
always be chosen to be a best nonnegative rank-one approximation, and that the
set of nonnegative tensors with non-unique best rank-one approximations form an
algebraic hypersurface. We show that the last part holds true more generally
for real tensors and thereby determine a polynomial equation so that a real or
nonnegative tensor which does not satisfy this equation is guaranteed to have a
unique best rank-one approximation. We also establish an analogue for real or
nonnegative symmetric tensors. In addition, we prove a singular vector variant
of the Perron--Frobenius Theorem for positive tensors and apply it to show that
a best nonnegative rank-r approximation of a positive tensor can never be
obtained by deflation. As an aside, we verify that the Euclidean distance (ED)
discriminants of the Segre variety and the Veronese variety are hypersurfaces
and give defining equations of these ED discriminants
On best rank one approximation of tensors
In this paper we suggest a new algorithm for the computation of a best rank
one approximation of tensors, called alternating singular value decomposition.
This method is based on the computation of maximal singular values and the
corresponding singular vectors of matrices. We also introduce a modification
for this method and the alternating least squares method, which ensures that
alternating iterations will always converge to a semi-maximal point. (A
critical point in several vector variables is semi-maximal if it is maximal
with respect to each vector variable, while other vector variables are kept
fixed.) We present several numerical examples that illustrate the computational
performance of the new method in comparison to the alternating least square
method.Comment: 17 pages and 6 figure
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