Multi-way PLS regression: Monotony convergence of tri-linear PLS2 and optimality of parameters

•It is shown that the tri-linear PLS2 procedure is convergent.•The sequences generated by the tri-linear PLS2 can be described as increasing or decreasing two specific criteria.•A hidden tensor is described allowing tri-linear PLS2 to search its best rank one approximation.•A link between multi-way PLS regression and the well-known PARAFAC model is highlighted.

The tri-linear PLS2 iterative procedure, an algorithm pertaining to the NIPALS framework, is considered. It was previously proposed as a first stage to estimate parameters of the multi-way PLS regression method. It is shown that the tri-linear PLS2 procedure is convergent. The procedure generates a sequence of parameters (scores and loadings), which can be described as increasing or decreasing two specific criteria. Furthermore, a hidden tensor is described allowing tri-linear PLS2 to search its best rank-one approximation. This tensor highlights the link between multi-way PLS regression and the well-known PARAFAC model. The parameters of the multi-way PLS regression method can be computed using three alternative procedures.