A Topology Layer for Machine Learning

TL;DR

This paper introduces a differentiable topology layer utilizing persistent homology for machine learning, enabling regularization, topological priors in generative models, and adversarial attacks, thus broadening the application of topological data analysis in deep learning.

Contribution

It presents a novel differentiable topology layer based on persistent homology, with three applications including regularization, topological loss in generative models, and adversarial attacks.

Findings

The topology layer can regularize data and model weights.

It allows incorporating topological priors into deep generative networks.

The layer enables topological adversarial attacks on deep networks.

Abstract

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code (www.github.com/bruel-gabrielsson/TopologyLayer) is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.