Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to distribute, it is hard to generalize to multifiltrations and is computationally prohibitive for big data-sets. In this paper we study the concept of Euler Characteristics Curves, for one parameter filtrations and Euler Characteristic Profiles, for multi-parameter filtrations. While being a weaker invariant in one dimension, we show that Euler Characteristic based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations and practical applicability for big data problems. In addition we show that the Euler Curves and Profiles enjoys certain type of stability which makes them robust tool in data analysis. Lastly, to show their practical applicability, multiple use-cases are considered.

翻译：拓扑数据分析工具可提供稳定的摘要，用于描述所考虑数据的形状。持久同调作为最标准且被广泛研究的数据摘要方法，存在若干局限性：其计算难以分布式实现，难以推广至多过滤情形，且对于大数据集而言计算代价过高。本文研究了单参数过滤的欧拉特征曲线与多参数过滤的欧拉特征剖面概念。尽管在一维情况下该不变量较弱，但我们证明基于欧拉特征的方法不存在持久同调的某些缺陷；我们提出了高效的分布式计算算法、向多过滤的推广方法，以及在大数据问题中的实际应用可行性。此外，我们证明欧拉曲线与剖面具有特定类型的稳定性，使其成为数据分析中的稳健工具。最后，通过多个用例验证了其实用性。