# Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology

## TL;DR

This paper introduces neural persistence, a novel topological complexity measure for neural networks that captures structural properties and can guide training efficiency improvements.

## Contribution

We propose neural persistence as a new complexity measure based on algebraic topology, enabling structural analysis and training optimization of neural networks.

## Key findings

- Neural persistence reflects effects of dropout and batch normalization.
- It can be used as a stopping criterion to reduce training time.
- It achieves comparable accuracy to traditional early stopping methods.

## Abstract

While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In this work, we propose neural persistence, a complexity measure for neural network architectures based on topological data analysis on weighted stratified graphs. To demonstrate the usefulness of our approach, we show that neural persistence reflects best practices developed in the deep learning community such as dropout and batch normalization. Moreover, we derive a neural persistence-based stopping criterion that shortens the training process while achieving comparable accuracies as early stopping based on validation loss.