# Latent Space Topology Evolution in Multilayer Perceptrons

arXiv: https://arxiv.org/abs/2506.01569 · HTML: https://mlsanity.com/paper/2506.01569/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 9 of 9)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Deep, Skinny Neural Networks are not Universal Approximators](https://mlsanity.com/paper/1810.00393/) (2018)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/2311.04592/) (2023)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Barcodes of Towers and a Streaming Algorithm for Persistent Homology](https://mlsanity.com/paper/1701.02208/) (2017)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)

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

- [Topological Attribution Distance (TAD): Revealing Segment-Level RAG Influence on LLM Output Geometry for Incident Log Analysis](https://mlsanity.com/paper/2608.16775/) (2026)

## Abstract (arXiv)

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.