On some limitations of probabilistic models for dimension-reduction: illustration in the case of one particular probabilistic formulation of PLS
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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).
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