  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.
