# Topological Simplification in Predictive Coding Networks

arXiv: https://arxiv.org/abs/2608.02816 · HTML: https://mlsanity.com/paper/2608.02816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 22 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Benchmarking Predictive Coding Networks -- Made Simple](https://mlsanity.com/paper/2407.01163/) (2024)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [Topological Measurement of Deep Neural Networks Using Persistent Homology](https://mlsanity.com/paper/2106.03016/) (2021)
- [On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/2311.04592/) (2023)
- [Introduction to Predictive Coding Networks for Machine Learning](https://mlsanity.com/paper/2506.06332/) (2025)

## Abstract (arXiv)

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 (≥ 99.9% test accuracy) and on MNIST (≥ 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 unicodex1D70C ∈ [0.72, 0.79] across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation (unicodex1D70C = -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.