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

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. To enhance the accuracy of such machine learning methods, it is known to be effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we need to choose a filtration for the point clouds, an increasing sequence of spaces. Because 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 theoretically show a finite-dimensional approximation result that justifies our architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.