On some limitations of probabilistic models for dimension‐reduction: illustration in the case of probabilistic formulations of Partial Least Squares

Partial Least Squares (PLS) refer to a class of dimension-reduction techniques aiming at the identification two sets components with maximal covariance, model relationship between observed variables $x\in\mathbb{R}^p$ and $y\in\mathbb{R}^q$, $p\geq 1, q\geq 1$. Probabilistic formulations have recently been proposed for several versions PLS. Focusing first on probabilistic formulation PLS-SVD by el Bouhaddani et al., we establish that constraints their parameters are too restrictive define particular distributions $(x,y)$, under which covariance (solutions PLS-SVD) also necessarily respective variances principal analyses $x$ $y$, respectively). We propose an alternative PLS-SVD, no longer restricted these distributions. then present numerical illustrations limitation original al. briefly discuss similar limitations in another latent variable dimension-reduction.