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Gaussian Regularized Sliced Inverse Regression

机译：高斯正则化切片逆回归

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摘要

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyper-spectral images are provided.
机译：切片逆回归（SIR）是减少高维回归问题中尺寸的有效方法。但是，原始方法需要对预测变量协方差矩阵进行求逆。如果这些预测变量之间存在共线性或与维数相比样本量较小，则无法进行反演，必须使用正则化技术。我们的方法基于库克（R.D. Cook）的Fisher讲座，该讲座表明SIR轴可以解释为逆回归问题的解决方案。我们建议在逆回归问题的未知参数上引入高斯先验分布，以规范其估计。我们表明，一些现有的SIR正则化可以进入我们的框架，从而使人们对这些方法有全局的了解。提出了三个新的先验，从而导致SIR方法的新正则化。提供了模拟数据的比较以及从高光谱图像估计火星表面物理性质的应用。

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