# POWER METHOD FOR THE MAXIMAL -SINGULAR VALUE OF NON-NEGATIVE TENSORS

by Antoine Gautier and Matthias Hein

## 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 p_{i}-norm on and is the multi-linear form associated to the tensor , i.e.

with .

**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

with .

**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 .

## CONVERGENCE GUARANTEE

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

Example:

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

such that

Example:

with 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.

## DOWNLOAD AND LICENSE

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

arXiv:1503.01273

if you find this code useful for your research.