通过线性化压缩特征，解释高维回归系数如何逼近非线性响应。
Interpretation of High-Dimensional Regression Coefficients by Comparison with Linearized Compressing Features
- 用线性化方法生成压缩特征系数，与正则化路径中系数对比。
- 在电池寿命预测中，单个非线性压缩特征可有效近似复杂响应。
- 揭示正则化如何影响系数形状以捕捉局部非线性结构，适合建模专家。
线性回归常被视为可解释，但在高维数据下面临挑战。本文聚焦于理解线性回归如何近似高维功能数据中的非线性响应，动机来自锂离子电池循环寿命预测。我们提出一种线性化方法，推导压缩特征的系数，并将其与回归解路径中最近的系数进行比较。在电池数据案例研究中，使用一个从 $\mathbb{R}^p \to \mathbb{R}$ 映射的非线性压缩特征 $g$ 构造合成响应 $\mathbf{y} \in \mathbb{R}$。该统一视角有助于理解：(1) 在高度正则化域中，回归系数如何被塑造及其与线性化特征系数的关系；(2) 正则化程度变化时，系数形状如何演化，从而利用局部结构逼近非线性响应。
原文摘要 · Abstract (English)
Linear regression is often deemed inherently interpretable; however, challenges arise for high-dimensional data. We focus on further understanding how linear regression approximates nonlinear responses from high-dimensional functional data, motivated by predicting cycle life for lithium-ion batteries. We develop a linearization method to derive feature coefficients, which we compare with the closest regression coefficients of the path of regression solutions. We showcase the methods on battery data case studies where a single nonlinear compressing feature, $g\colon \mathbb{R}^p \to \mathbb{R}$, is used to construct a synthetic response, $\mathbf{y} \in \mathbb{R}$. This unifying view of linear regression and compressing features for high-dimensional functional data helps to understand (1) how regression coefficients are shaped in the highly regularized domain and how they relate to linearized feature coefficients and (2) how the shape of regression coefficients changes as a function of regularization to approximate nonlinear responses by exploiting local structures.
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