The limits of interpretability in multiple linear regression

2026-06-16 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Artificial Intelligence & Machine Learning, Physical Sciences & Chemistry, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

A study by Sharma, Coslovich, and Ozawa investigates the limitations of interpretability in multiple linear regression (MLR) models, particularly when input features exhibit strong correlations, known as multicollinearity. While MLR is often considered interpretable due to its explicit weighted sums of features, the authors demonstrate that multicollinearity causes learned weights to show large dataset-to-dataset fluctuations and oscillatory patterns across physically similar features, hindering meaningful interpretation. Their theoretical analysis, based on the eigenmodes of the feature correlation matrix, reveals that small-eigenvalue modes amplify these fluctuations and generate non-meaningful oscillatory patterns. Numerical tests on physics datasets, including superconductivity (81 features, 21263 data points) and glassy-dynamics (60 features, 40000 data points), confirm these findings. Ridge regularization is shown to suppress these unstable modes, though the resulting weights still require cautious interpretation, as their patterns can change significantly with regularization strength even when predictive performance remains stable.

Key takeaway

For Research Scientists and Data Scientists building linear models, recognize that high predictive accuracy does not equate to interpretable feature weights, especially with multicollinearity. You should routinely calculate the Variance Inflation Factor (VIF) for your features; if VIF values exceed 10, your OLS weights are likely unstable and misleading. While Ridge regression can stabilize weights and mitigate oscillations, carefully evaluate how regularization strength alpha impacts weight patterns, not just predictive performance, to ensure physically consistent interpretations.

Key insights

Multicollinearity in linear regression fundamentally limits weight interpretability by causing unstable and oscillatory feature contributions.

Principles

• Good prediction performance does not guarantee reliable weight interpretability.
• Multicollinearity amplifies weight fluctuations via small-eigenvalue modes.
• Ridge regularization stabilizes weights but requires careful interpretation.

Method

The study theoretically clarifies interpretability limits by analyzing the eigenmodes of the feature correlation matrix for Ordinary Least Squares (OLS) and Ridge regression.

In practice

• Use Variance Inflation Factor (VIF) to quantify multicollinearity severity.
• Apply Ridge regularization with alpha values around 10^-2 to 10^-1 to stabilize weights.

Topics

• Multiple Linear Regression
• Model Interpretability
• Multicollinearity
• Ridge Regression
• Feature Correlation Matrix
• Variance Inflation Factor

Best for: AI Scientist, Research Scientist, Data Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.