Shortcut learning in geometric knot classification

TL;DR

This paper investigates how machine learning models classify knots, revealing that they often rely on hidden non-topological features, and provides datasets and tools to improve topological classification accuracy.

Contribution

The authors identify non-topological features used by ML in knot classification and provide datasets and code to facilitate topologically faithful learning.

Findings

ML models exploit hidden geometric features in training data.

A new dataset helps distinguish topological from non-topological features.

Tools are provided to generate knot embeddings with fixed topology.

Abstract

Classifying the topology of closed curves is a central problem in low dimensional topology with applications beyond mathematics spanning protein folding, polymer physics and even magnetohydrodynamics. The central problem is how to determine whether two embeddings of a closed arc are equivalent under ambient isotopy. Given the striking ability of neural networks to solve complex classification tasks, it is therefore natural to ask if the knot classification problem can be tackled using Machine Learning (ML). In this paper, we investigate generic shortcut methods employed by ML to solve the knot classification challenge and specifically discover hidden non-topological features in training data generated through Molecular Dynamics simulations of polygonal knots that are used by ML to arrive to positive classifications results. We then provide a rigorous foundation for future attempts to tackle the knot classification challenge using ML by developing a publicly-available (i) dataset, that aims to remove the potential of non-topological feature classification and (ii) code, that can generate knot embeddings that faithfully explore chosen geometric state space with fixed knot topology. We expect that our work will accelerate the development of ML models that can solve complex geometric knot classification challenges.