Interpretation of High-Dimensional Regression Coefficients by Comparison with Linearized Compressing Features
This work addresses interpretability issues in high-dimensional regression for domain-specific applications like battery life prediction, but it is incremental as it builds on existing linearization and regularization techniques.
The paper tackles the challenge of interpreting linear regression coefficients for high-dimensional functional data, specifically in predicting lithium-ion battery cycle life, by developing a linearization method to compare feature coefficients with regression coefficients and analyzing their behavior under regularization.
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.