In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieve state-of-the-art performance in supervised tasks at a very low computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.

翻译：本文研究了拓扑数据分析中的欧拉示性数技术。对由数据构建的单纯复形族逐点计算欧拉示性数，可得到所谓的欧拉示性数轮廓。我们证明，这一简单描述符在极低计算成本下即可在监督任务中实现最先进的性能。受信号分析启发，我们计算了欧拉示性数轮廓的混合变换。这些积分变换将欧拉示性数技术与勒贝格积分相结合，为拓扑信号提供了高效压缩器，因此在无监督环境下展现出卓越表现。在定性层面，我们提出了大量关于欧拉轮廓及其混合变换所捕获拓扑与几何信息的启发式方法。最后，我们证明了这些描述符的稳定性结果以及随机环境下的渐近保证。