提出AffineLens框架，精确计算神经网络的分段线性区域结构。
AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks
- 基于输入域内超平面交集，逐层枚举分段线性区域。
- 可准确统计区域数量并生成可视化分区图，支持主流网络结构。
- 适合研究模型几何表达能力或需解释决策边界的场景。
分段仿射神经网络（PANNs）通过将输入-输出映射建模为连续分段仿射（CPA）函数，提供了理解神经网络表达能力的几何视角，其复杂度由仿射区域的数量、排列和形状决定。然而，现有可解释性与表达能力分析多依赖间接代理（如激活统计或理论上限），缺乏在真实架构和有界输入域下精确枚举与可视化所诱导区域划分的实际工具。本文提出AffineLens，一个统一框架，用于计算支撑PANNs的超平面排列与多面体结构。给定校准的（有界）输入多面体，AffineLens识别与该域相交的神经元诱导超平面，逐层枚举生成的仿射子区域，并返回可证明非空的最大CPA区域及其内部代表点。该框架还提供区域划分与决策边界的可视化，支持定性检查与定量区域计数。通过利用CPA网络在固定激活模式下的仿射约束特性，AffineLens适用于包括批归一化、池化、残差连接、多层感知机和卷积层在内的广泛现代组件。最后，我们使用AffineLens对网络架构的表达能力进行系统性实证研究，通过区域复杂度指标比较不同网络，揭示设计选择如何影响学习函数的几何结构。
原文摘要 · Abstract (English)
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.
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