Persistence-based operators in machine learning

TL;DR

This paper introduces persistence-based neural network layers that incorporate topological data analysis principles, enabling the integration of data symmetries and learnable parameters into neural architectures.

Contribution

It presents a novel class of neural network layers grounded in topological data analysis, facilitating symmetry-aware learning with learnable weights.

Findings

Persistence-based layers can encode data symmetries effectively.

They are compatible with existing neural architectures.

The approach enhances interpretability and data constraint adherence.

Abstract

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines fully aware of data constraints and symmetries. We introduce a class of persistence-based neural network layers. Persistence-based layers allow the users to easily inject knowledge about symmetries (equivariance) respected by the data, are equipped with learnable weights, and can be composed with state-of-the-art neural architectures.