Latent Space Topology Evolution in Multilayer Perceptrons

TL;DR

This paper presents a topological framework for analyzing how data representations evolve across layers in Multilayer Perceptrons, providing new insights into their internal organization and transformation of data topology.

Contribution

It introduces a novel topological approach using simplicial complexes and persistence analysis to interpret MLP internal representations and their evolution.

Findings

Identifies topological features associated with layer transformations

Reveals critical topological transitions in data processing

Detects redundant layers and interprets data organization

Abstract

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.