# AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks

arXiv: https://arxiv.org/abs/2605.06218 · HTML: https://mlsanity.com/paper/2605.06218/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 47 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Reverse-Engineering Deep ReLU Networks](https://mlsanity.com/paper/1910.00744/) (2019)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [The Geometry of Deep Networks: Power Diagram Subdivision](https://mlsanity.com/paper/1905.08443/) (2019)
- [Empirical Studies on the Properties of Linear Regions in Deep Neural Networks](https://mlsanity.com/paper/2001.01072/) (2020)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [Improved Bounds on Neural Complexity for Representing Piecewise Linear Functions](https://mlsanity.com/paper/2210.07236/) (2022)
- [On the Number of Regions of Piecewise Linear Neural Networks](https://mlsanity.com/paper/2206.08615/) (2022)
- [On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth](https://mlsanity.com/paper/2502.06283/) (2025)
- [Deep ReLU Networks Preserve Expected Length](https://mlsanity.com/paper/2102.10492/) (2021)
- [On the Expected Complexity of Maxout Networks](https://mlsanity.com/paper/2107.00379/) (2021)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Effects of Data Geometry in Early Deep Learning](https://mlsanity.com/paper/2301.00008/) (2022)
- [Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification](https://mlsanity.com/paper/2306.08805/) (2023)
- [Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks](https://mlsanity.com/paper/2206.00228/) (2022)
- [On the Local Complexity of Linear Regions in Deep ReLU Networks](https://mlsanity.com/paper/2412.18283/) (2024)
- [Piecewise Linear Neural Networks and Deep Learning](https://mlsanity.com/paper/2206.09149/) (2022)
- [Understanding the Evolution of Linear Regions in Deep Reinforcement Learning](https://mlsanity.com/paper/2210.13611/) (2022)
- [Fast and Exact Enumeration of Deep Networks Partitions Regions](https://mlsanity.com/paper/2401.11188/) (2024)
- [Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training](https://mlsanity.com/paper/2311.18022/) (2023)
- [Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Time](https://mlsanity.com/paper/2505.18023/) (2025)
- [The Evolution of the Interplay Between Input Distributions and Linear Regions in Networks](https://mlsanity.com/paper/2310.18725/) (2023)

## Built on by (most cited first, 1 of 1)

- [Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity](https://mlsanity.com/paper/2606.21687/) (2026)

## Abstract (arXiv)

Piecewise affine neural networks (PANNs) provide a principled geometric perspective on neural network expressivity by characterizing the input--output map as a continuous piecewise affine (CPA) function whose complexity is governed by the number, arrangement, and shapes of its affine regions. However, existing interpretability and expressivity analyses often rely on indirect proxies (e.g., activation statistics or theoretical upper bounds) and rarely offer practical, accurate tools for enumerating and visualizing the induced region partition under realistic architectures and bounded input domains. In this work, we present AffineLens, a unified framework for computing the hyperplane arrangements and polyhedral structures underlying PANNs. Given a calibrated (bounded) input polytope, AffineLens identifies the subset of neuron-induced hyperplanes that intersect the domain, enumerates the resulting affine sub-regions in a layer-wise manner, and returns provably non-empty maximal CPA regions together with interior representatives. The framework further provides visualizations of region partitioning and decision boundaries, enabling qualitative inspection alongside quantitative region counts. By exploiting the affine restriction property of CPA networks under fixed activation patterns, AffineLens supports a broad class of modern components, including batch normalization, pooling, residual connections, multilayer perceptrons, and convolutional layers. Finally, we use AffineLens to perform a systematic empirical study of architectural expressivity, comparing networks through region complexity metrics and revealing how design choices influence the geometry of learned functions.