On some limitations of probabilistic models for dimension-reduction: illustration in the case of one particular probabilistic formulation of PLS
share
Partial Least Squares (PLS) refer to a class of dimension-reduction techniques aiming at the identification of two sets of components with maximal covariance, in order to model the relationship between two sets of observed variables x∈R^p and y∈R^q, with p≥ 1, q≥ 1. El Bouhaddani et al. (2017) have recently proposed a probabilistic formulation of PLS. Under the constraints they consider for the parameters of their model, this latter can be seen as a probabilistic formulation of one version of PLS, namely the PLS-SVD. However, we establish that these constraints are too restrictive as they define a very particular subset of distributions for (x,y) under which, roughly speaking, components with maximal covariance (solutions of PLS-SVD), are also necessarily of respective maximal variances (solutions of the principal components analyses of x and y, respectively). Then, we propose a simple extension of el Bouhaddani et al.'s model, which corresponds to a more general probabilistic formulation of PLS-SVD, and which is no longer restricted to these particular distributions. We present numerical examples to illustrate the limitations of the original model of el Bouhaddani et al. (2017).
READ FULL TEXT