λ(f): A Linearity Score for Neural Network Interpretability

This repository contains the code and datasets for the paper:

"Faithful to What?" On the Limits of Fidelity-Based Explanations
Jackson Eshbaugh, Department of Computer Science, Lafayette College, January 2026

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Overview

Neural network outputs can often be well-approximated by linear models—but what does that tell us?

This project introduces the linearity score λ(f), a simple diagnostic that measures how well a regression network’s input–output behavior can be approximated by a linear model. We show that high linear fidelity to a network does not necessarily imply high task accuracy, highlighting a gap between being faithful to a model and being faithful to the underlying data-generating signal.

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What is λ(f)?

Let $f$ be a trained regression network, and let $\mathcal{L}$ be the space of affine functions. Define:

$ \lambda(f) := R^2(f, g^) = 1 - \frac{\mathbb{E}[(f(x) - g^(x))^2]}{\text{Var}(f(x))} $

where $ g^* = \arg\min_{g \in \mathcal{L}} \mathbb{E}[(f(x) - g(x))^2]$.

In other words, $\lambda(f)$ measures how well a linear model can mimic the predictions of a trained neural network. Unlike typical $R^2$, this score is not about matching the ground truth—it’s about measuring how linearly decodable the function learned by the network is from the input space.

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Reproducing Results

All experiments and visualizations in the paper are contained in:

📓 lambda_linearity_score.ipynb

The notebook is fully self-contained and organized into:

• A reusable experimental framework
• Four datasets (synthetic + 2 real-world)
• Plots and tabulated results

To use $\lambda(f)$ on your own data, modify the preprocessing and build_network() function and rerun the provided pipeline.

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Datasets Used

• Synthetic: $y = x \cdot \sin(x) + \varepsilon$, where $\varepsilon \sim \mathcal{N}(0, \sigma^2)$
• Medical Cost Personal Dataset (Kaggle)
• California Housing (scikit-learn)

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Requirements

To run the notebook, install the following Python packages:

pip install tensorflow scikit-learn matplotlib pandas seaborn kagglehub

Tested with Python 3.11.

Acknowledgements

Special thanks to Professor Jorge Silveyra for the early discussions that helped spark this project.

© 2026 Jackson Eshbaugh • Lafayette College • Released under the MIT License.