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.

翻译：点云的机器学习近年来受到广泛关注，在形状识别、材料科学等多个领域具有重要应用。为了提升这类机器学习方法的精度，融合全局拓扑特征（通常通过持续同调提取）被证明是有效手段。在计算点云的持续同调时，需要定义点云的过滤——即一个递增的空间序列。由于融合持续同调的机器学习方法的性能高度依赖过滤的选择，需根据数据特征和任务目标进行调节。本文提出一种利用神经网络自适应学习过滤机制的框架。为使生成的持续同调具有等距不变性，我们设计了具有该不变性的神经网络架构。此外，我们从理论上证明了该架构的有限维近似定理。实验结果表明，该框架在多项分类任务中展现出显著效果。