Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.

动力学过程的有意义的低维表示对于更好地理解复杂系统的机制至关重要，从音乐创作到生物学和人工智能的学习，这些复杂系统都是如此。我们建议通过使用基于持久同源性的方法，通过考虑随时间变化的系统的几何和拓扑特性来描述随时间变化的系统。在静态情况下，持久性同源性允许人们提供与连续功能配对的流形的表示，这些流形是称为持久性图的点和线的多组集合。想法是将可变几何空间的变化作为持久性图的时间序列进行指纹识别，然后通过使用动态时间扭曲来比较此类时间序列。作为应用，吨数。此后，我们使用这种基于时间的表示形式，根据其风格自动对三个乐曲集合进行分类，并讨论了用于分析不同音乐体裁的最佳时间粒度。