Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds

TL;DR

This paper introduces an adaptive neural network framework for learning filtration functions in persistent homology, improving topological feature extraction from point clouds for classification tasks.

Contribution

It proposes a neural network architecture that learns filtration functions adaptively and is isometry-invariant, with theoretical justification and demonstrated effectiveness.

Findings

Improved classification accuracy on point cloud datasets.

Neural network effectively learns filtration functions tailored to data.

Theoretical validation of finite-dimensional approximation of filtrations.

Abstract

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.