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

中文翻译：

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关于降维概率模型的一些限制：以偏最小二乘的概率公式为例

偏最小二乘法（PLS）是一类降维技术，旨在识别具有最大协方差的两组分量，对两组观测变量之间的关系进行建模x ∈Rp$X\in {ℝ}^p$$x\in {ℝ}^p$和是的∈Rq$\mathrm{是的}\in {ℝ}^q$$y\in {ℝ}^q$， 和p≥1 , q _ _≥1 _$p\ge 1,q\ge 1$$p\ge 1,q\ge 1$. 最近已经为 PLS 的几个版本提出了概率公式。首先关注 el Bouhaddani 等人提出的 PLS-SVD 的概率公式，我们确定对其模型参数的约束过于严格，并定义了特定的分布( x , y)$(X,\mathrm{是的})$$(x,y)$，其中具有最大协方差的分量（PLS-SVD 的解）也必然具有各自的最大方差（分别为x和y的主成分分析的解）。我们提出了 PLS-SVD 的另一种概率公式，不再局限于这些特定的分布。然后，我们展示了 el Bouhaddani 等人的原始模型的局限性的数值说明。我们还简要讨论了另一个用于降维的潜在变量模型中的类似限制。