Equivalence of state equations from different methods in high-dimensional regression

State equations (SEs) were first introduced in the approximate message passing process to describe the mean square error in compressed sensing. Since then a set of state equations has appeared in studies of logistic regression, robust estimator, and other high-dimensional statistics problems. Recently, a convex Gaussian min-max theorem approach was proposed to study high-dimensional statistic problems accompanying with another set of different state equations. This article 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.