关于降维的概率模型的某些局限性：偏最小二乘概率公式的例证
On some limitations of probabilistic models for dimension-reduction: Illustration in the case of probabilistic formulations of partial least squares
摘要：
偏最小二乘（PLS）是一类降维技术，旨在识别具有最大协方差的两组分量，以利用$p\geq 1, q\geq 1$对两组观测变量$x\in\mathbb{R}^p$和$y\in\mathbb{R}^q$之间的关系进行建模。最近已经提出了几个版本的PLS的概率公式。首先关注el Bouhaddani等人提出的PLS-SVD的概率公式，我们建立了对它们的模型参数的约束过于严格，并且为$(x,y)$定义了特定的分布，在该分布下，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（分别是$x$和$y$的主成分分析的解）。我们提出了一种替代的概率公式PLS-SVD，不再局限于这些特定的分布。然后，我们提出了数值说明的限制，EL Bouhaddani等人的原始模型。我们还简要讨论了类似的限制，在另一个潜在的变量模型降维。
英文摘要：
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 $x\in\mathbb{R}^p$ and $y\in\mathbb{R}^q$, with $p\geq 1, q\geq 1$. 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 $(x,y)$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
