Unreduced Persistence Diagrams for Topological Machine Learning

2026-06-18 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

Parker B. Edwards introduces methods to generate topological feature vectors from unreduced boundary matrices, challenging the conventional reliance on fully-reduced persistence diagrams (FR PDs) in topological machine learning (TML) pipelines. The study explores three types of unreduced PDs: non-negative betas (NNB), apparent pair (AP), and naive low-ones (L1). Experiments across synthetic shape classification, Fashion MNIST image classification, and brain artery tree regression tasks demonstrate that models trained on unreduced PDs can perform comparably to, and in some cases, outperform those using FR PDs. Notably, L1 diagrams clearly outperformed FR diagrams in Fashion MNIST classification, achieving 78% accuracy versus 66% for simplified models, and 91% versus 84% for specific classes. This suggests that utilizing unreduced boundary matrix information could offer computational cost savings by avoiding the demanding full reduction step, despite potentially longer vectorization times for larger unreduced diagrams.

Key takeaway

For Machine Learning Engineers optimizing topological data analysis pipelines, you should investigate integrating unreduced persistence diagrams. Specifically, consider L1 diagrams, which demonstrated comparable or superior performance to fully-reduced diagrams in tasks like Fashion MNIST classification. This approach can potentially reduce the significant computational burden of boundary matrix reduction, allowing for more efficient model training. Benchmark the vectorization costs of larger unreduced diagrams against the savings from skipping reduction.

Key insights

Unreduced persistence diagrams can match or exceed fully-reduced PD performance in topological machine learning, potentially saving computation.

Principles

• Full boundary matrix reduction may be computationally wasteful.
• Unreduced PDs can encode distinct, useful information.
• Performance varies by PD type and task.

Method

The study compared four diagram constructions (FR, NNB, AP, L1) using a PH/ML pipeline template, vectorizing PDs with persistence images or Adcock-Carlsson coordinates, and training random forest classifiers/regressors.

In practice

• Consider L1 diagrams for Fashion MNIST-like image classification.
• Explore unreduced PDs to reduce persistent homology computation.
• Benchmark computational trade-offs for vectorizing larger diagrams.

Topics

• Topological Data Analysis
• Persistent Homology
• Unreduced Persistence Diagrams
• Machine Learning Pipelines
• Fashion MNIST
• Computational Topology

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.