Multi-way PLS for discrimination: Compact form equivalent to the tri-linear PLS2 procedure and its monotony convergence

• A compact formulation of the tri-linear PLS2 procedure was proposed.
• Equivalence to tri-linear PLS2 was demonstrated.
• Convergence criteria were proved.

This paper focuses on multi-way PLS for discrimination. Emphasis was placed on the computation of parameters (so-called scores and loadings) using an iterative procedure called tri-linear PLS2. In the context of discrimination, this algorithm is applied to a dummy matrix representing group membership, and according to its specific formalism, the tri-linear PLS2 procedure offers the possibility of simplification. The purpose of this paper is to introduce a compact formulation of the tri-linear PLS2 procedure adapted for a discrimination setting. A property of this variant that allows formally establishing its convergence and, by extension, the convergence of the tri-linear PLS2 will be demonstrated. The potential of this compact form will be illustrated with simulated examples.