The paper shows that the application of the fixed-effect multiple linear
regression model to an overparameterized dataset is equivalent to fitting the
data with a hyper-curve parameterized by a single scalar parameter. This
equivalence allows for a predictor-focused approach, where each predictor is
described by a function of the chosen parameter. It is proven that a linear
model will produce exact predictions even in the presence of nonlinear
dependencies that violate the model assumptions. Parameterization in terms of
the dependent variable and the monomial basis in the predictor function space
are applied here to both synthetic and experimental data. The hyper-curve
approach is especially suited for the regularization of problems with noise in
predictor variables and can be used to remove noisy and "improper" predictors
from the model.

该论文表明，将固定效应多重线性回归模型应用于过度参数化的数据集等同于用单一标量参数对数据进行拟合的超曲线参数化。通过这种等价性，可以采用以预测变量为中心的方法，其中每个预测变量都由所选参数的函数来描述。证明了即使存在违反模型假设的非线性依赖关系，线性模型也能够产生精确预测。在此处应用了以因变量和预测变量函数空间的单项基为参数化的方法来处理合成和实验数据。超曲线方法特别适用于处理预测变量中的噪声和 “不合适” 预测变量的正则化问题。