Topological Measurement of Deep Neural Networks Using Persistent Homology

TL;DR

This paper introduces a topological data analysis approach using persistent homology to analyze the complex inner representations of deep neural networks, providing insights into neuron interactions and problem difficulty.

Contribution

It applies persistent homology to DNNs to reveal their internal structure, a novel use of topological data analysis in neural network interpretability.

Findings

PH reflects neuron excess in DNNs

PH correlates with problem difficulty

Method effective on FCNs and CNNs trained on MNIST and CIFAR-10

Abstract

The inner representation of deep neural networks (DNNs) is indecipherable, which makes it difficult to tune DNN models, control their training process, and interpret their outputs. In this paper, we propose a novel approach to investigate the inner representation of DNNs through topological data analysis (TDA). Persistent homology (PH), one of the outstanding methods in TDA, was employed for investigating the complexities of trained DNNs. We constructed clique complexes on trained DNNs and calculated the one-dimensional PH of DNNs. The PH reveals the combinational effects of multiple neurons in DNNs at different resolutions, which is difficult to be captured without using PH. Evaluations were conducted using fully connected networks (FCNs) and networks combining FCNs and convolutional neural networks (CNNs) trained on the MNIST and CIFAR-10 data sets. Evaluation results demonstrate that the PH of DNNs reflects both the excess of neurons and problem difficulty, making PH one of the prominent methods for investigating the inner representation of DNNs.