用拓扑方法追踪MLP中数据如何逐层变形，揭示分类背后的几何逻辑。
Latent Space Topology Evolution in Multilayer Perceptrons
- 构建单纯复形序列，追踪数据在各层的拓扑演变
- 发现隐空间线性可分性与神经元连接断点相关
- 可视化数据流轨迹，定位冗余层与关键转变点
本文提出一种用于解析多层感知机（MLPs）内部表示的拓扑框架。我们构建了一个单纯复形塔，即由单纯映射连接的单纯复形序列，以捕捉数据拓扑随网络层数的变化过程。该方法支持双持久性分析：层持久性跟踪每层内拓扑特征在不同尺度下的稳定性，而MLP持久性揭示这些特征在网络中的演化路径。我们证明了拓扑描述符的稳定性定理，并建立线性可分性与神经复形中分离连通分支之间的关联。为提升实用性，我们设计了组合算法计算MLP持久性，并引入基于轨迹的可视化方法，追踪数据在网络中的流动。在合成数据和真实医疗数据上的实验表明，该方法能识别冗余层、揭示关键拓扑转变，并提供可解释的见解，说明MLP如何逐步组织数据以实现分类。
原文摘要 · Abstract (English)
This paper introduces a topological framework for interpreting the internal representations of Multilayer Perceptrons (MLPs). We construct a simplicial tower, a sequence of simplicial complexes connected by simplicial maps, that captures how data topology evolves across network layers. Our approach enables bi-persistence analysis: layer persistence tracks topological features within each layer across scales, while MLP persistence reveals how these features transform through the network. We prove stability theorems for our topological descriptors and establish that linear separability in latent spaces is related to disconnected components in the nerve complexes. To make our framework practical, we develop a combinatorial algorithm for computing MLP persistence and introduce trajectory-based visualisations that track data flow through the network. Experiments on synthetic and real-world medical data demonstrate our method's ability to identify redundant layers, reveal critical topological transitions, and provide interpretable insights into how MLPs progressively organise data for classification.
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