In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped with a multiparameter filtration. Multiparameter persistence has become one of the most active areas of research within TDA, with exciting progress on several fronts. In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future.
翻译:在地形数据分析(TDA)中,人们往往通过建造一个过滤的表层空间来研究数据形状,然后用持续的同系物来研究其结构,然而,一个单一的过滤空间往往不能充分捕捉数据中感兴趣的结构,而一个则导致考虑多参数的持久性,它与一个配备多参数过滤器的空间的数据相联系,多参数的持久性已成为TDA中最活跃的研究领域之一,在几个方面取得了令人振奋的进展。在本篇文章中,我们引入了多参数的持久性,并调查了最近取得的一些进展,重点是有可能在不久的将来产生实际应用的想法。</s>