Existence of the best low-rank tensor approximate
Résumé
The so-called Canonical Polyadic (CP) tensor decomposition, sometimes referred to as Parafac, is now widely utilized in Chemometrics, and in particular in the analysis of Fluorescence Emission Excitation Matrices (FEEM), if a set of several such matrices is available. Four messages are put forward: (i) the parameterization is important, and should fix the scaling indeterminacy; (ii) Kruskal's lemma guarantees uniqueness of a rank-R decomposition, but is not adequate for nonnegative tensors, nor for low-rank approximations; (iii) the best low-rank approximate does not always exist, but it does if the nonnegativity constraint is imposed; (iv) hence it is mandatory to use numerical algorithms imposing the nonnegativity constraint, otherwise spurious components may appear.