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

Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, Karsten Borgwardt

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.

Keywords: Algebraic topology, persistent homology, network complexity, neural network

TL;DR: We develop a new topological complexity measure for deep neural networks and demonstrate that it captures their salient properties.

Code: [ BorgwardtLab/Neural-Persistence](https://github.com/BorgwardtLab/Neural-Persistence) + [ 1 community implementation](https://paperswithcode.com/paper/?openreview=ByxkijC5FQ)

Data: [Fashion-MNIST](https://paperswithcode.com/dataset/fashion-mnist)