POWER METHOD FOR THE MAXIMAL -SINGULAR VALUE OF NON-NEGATIVE TENSORS
TENSOR PROJECTIVE NORM
Given a tensor of order m and , we propose in our paper, a power method type algorithm for computing a maximizer of the following problem:
where denotes the usual pi-norm on and is the multi-linear form associated to the tensor , i.e.
Matrix case: When the tensor is of order m = 2, i.e. is a matrix, the maximization problem above reduces to the so-called matrix p, q-norm, namely
Remark 1. Provided that the convergence assumptions are fulfilled, the algorithm produces two monotonic sequences such that
and thus can also be used to simply obtain an upper bound on .
The method is guaranteed to converge to a global maximizer of the optimization problem above if:
a) is non-negative, i.e. for all .
b) and there exists such that
c) The tensor is weakly irreducible. That is, the undirected m-partite graph G(T) = (V,E) is connected where
is the disjoint union of and
Matrix case: When the tensor is a matrix, then the three conditions above reduce to:
a) A is non-negative, i.e. for every .
b) and .
c) A is weakly irreducible (see definition above).
Example: If , then the undirected graph G(A) is given by
and therefore A is weakly irreducible because G(A) is connected.
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Please include a reference to our preprint
- A. Gautier and M. Hein
Tensor norm and maximal singular vectors of non-negative tensors - a Perron-Frobenius theorem, a
Collatz-Wielandt characterization and a generalized power method
if you find this code useful for your research.