[Paper Review] A Topology Layer for Machine Learning

Rickard Brüel‐Gabrielsson, Bradley J. Nelson+4 more|arXiv (Cornell University)|May 29, 2019

Topological and Geometric Data Analysis35 references46 citations

TL;DR

A differentiable topology layer computes persistent homology for level set and edge-based filtrations, enabling regularization, topological priors in generative models, and topological adversarial attacks.

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.

Motivation & Objective

• Introduce a differentiable persistence layer for level-set and edge-based filtrations applicable to Euclidean data, images, and 3D data.
• Demonstrate regularization by penalizing topological features directly on model weights or outputs.
• Incorporate topological priors into deep generative networks to improve generated data quality.
• Showcase topological adversarial attacks that manipulate persistence features to affect classifications.

Proposed method

• Compute persistent homology via level-set and edge-based filtrations with a differentiable framework.
• Provide a PyTorch extension enabling backpropagation through persistence diagrams.
• Define loss functions on persistence diagrams (e.g., E(p,q,i0;PDk) and Wasserstein) for gradient-based optimization.
• Map birth-death events to filtration cells to enable gradient computation with respect to inputs or parameters.
• Apply topological losses to regularize linear models and to refine GAN-generated images and 3D voxel data.
• Demonstrate adversarial attacks by optimizing inputs to alter topological features and classifications.

Experimental results

Research questions

• RQ1 Can a differentiable persistence layer be integrated into mainstream gradient-based learning pipelines?
• RQ2 How do topological penalties influence regularization and feature clustering compared to traditional norms?
• RQ3 What is the effect of topological priors on the quality of images and 3D voxel data generated by GANs?
• RQ4 Can persistence-based features enable effective topological adversarial attacks and reveal model vulnerabilities?

Key findings

• A differentiable topology layer enables backpropagation through persistence diagrams for common filtrations.
• Topological regularization can encourage clustering of model weights or reduce the number of local maxima.
• Topological priors in generative models can improve generated image quality and alter latent space interpolations to favor topology-consistent morphing.
• Topological adversarial attacks can manipulate persistence features to induce misclassification, with varying effectiveness across model types.
• Empirical results indicate measurable improvements in generator evaluation metrics when using topology-aware training.
• The approach supports 0D and 1D persistence with both level-set and edge-based filtrations across 2D images and 3D voxel data.