发表机构
University of Southern California（南加州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究用分层持续同调分析，发现预测编码网络（PCNs）的模型规模、简化深度与重构误差、架构类型影响其连通分量合并时机，揭示了PCNs压缩-重构权衡的相关规律。
AI 中文摘要
我们采用定量的分层持续同调分析，研究受神经启发的双向架构——预测编码网络（PCNs）中学习到的表示的拓扑结构。我们在合成分类数据集（测试准确率≥99.9%）和MNIST数据集（测试准确率≥95%）上训练性能良好的PCNs，测量不同架构和激活函数下各层拓扑特征的变化。研究发现，较小的PCNs（模型规模以隐藏层宽度之和衡量）比较大的模型更早在各层合并连通分量（不同激活函数下的斯皮尔曼相关系数ρ∈[0.72,0.79]）；还观察到简化发生的深度与重构误差呈强负相关（ρ=-0.58），即简化发生越晚的架构重构效果越好。最后，对不同架构和激活函数的种子级自举比较显示，PCNs合并连通分量的时间始终比匹配的多层感知机（MLPs）晚，平均相差3.6层。这些结果表明，持续同调为PCNs中的压缩-重构权衡提供了有用的定量视角，且模型容量与预测编码推理的循环双向动态共同决定了该权衡在各层的解决时机。
英文摘要
We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression--reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.
CommentsAccepted to the 2nd Annual Conference on Topology, Algebra, and Geometry in Data Science (TAG-DS 2026); to appear in Proceedings of Machine Learning Research
