论文标题
高维回归中不同方法的状态方程的等效性
Equivalence of state equations from different methods in High-dimensional Regression
论文作者
论文摘要
首先在近似消息传递（AMP）中引入状态方程（SES），以描述压缩感应中的均方根误差（MSE）。从那时起，一组状态方程出现在逻辑回归，健壮估计器和其他高维统计问题的研究中。最近，提出了一种凸高斯最低 - 最大定理（CGMT）方法来研究与另一组不同状态方程相伴的高维统计问题。本文在这些方法上提供了一个统一的观点，并显示了其还原形式的等效性，这导致所得的SE基本上是等效的，并且可以通过参数转换将其转换为相同的表达式。结合了这些结果，我们表明这些不同的状态方程是从几种等效的还原形式得出的。我们认为，这种等价将阐明在高维统计中发现更深层次的结构。
State equations (SEs) were firstly introduced in the approximate message passing (AMP) to describe the mean square error (MSE) in compressed sensing. Since then a set of state equations have appeared in studies of logistic regression, robust estimator and other high-dimensional statistics problems. Recently, a convex Gaussian min-max theorem (CGMT) approach was proposed to study high-dimensional statistic problems accompanying with another set of different state equations. This paper provides a uniform viewpoint on these methods and shows the equivalence of their reduction forms, which causes that the resulting SEs are essentially equivalent and can be converted into the same expression through parameter transformations. Combining these results, we show that these different state equations are derived from several equivalent reduction forms. We believe that this equivalence will shed light on discovering a deeper structure in high-dimensional statistics.
