一个计算效率高的持久性图向量表示框架
A computationally efficient framework for vector representation of persistence diagrams
摘要：
在拓扑数据分析中，量化数据形状的常用方法是使用持久性图（PD）。PD是使用代数拓扑工具计算的$\mathbb{R}^2$中的多点集。然而，这种多集合结构限制了PD在应用中的效用。因此，近年来的努力已经指向从PD中提取信息丰富且有效的摘要，以拓宽其用于机器学习任务的范围。我们提出了一个计算效率高的框架转换成一个矢量的PD在$\mathbb{R}^n$，称为矢量化持久块（VPB）。我们表明，我们的表示具有许多基于向量的摘要所需的属性，如相对于输入噪声的稳定性，低计算成本和灵活性。通过仿真研究，我们证明了VPB的有效性，在各种学习任务，即聚类，分类和变点检测的性能和计算成本。
英文摘要：
In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in $\mathbb{R}^2$ computed using tools of algebraic topology. However, this multi-set structure limits the utility of PDs in applications. Therefore, in recent years efforts have been directed towards extracting informative and efficient summaries from PDs to broaden the scope of their use for machine learning tasks. We propose a computationally efficient framework to convert a PD into a vector in $\mathbb{R}^n$, called a vectorized persistence block (VPB). We show that our representation possesses many of the desired properties of vector-based summaries such as stability with respect to input noise, low computational cost and flexibility. Through simulation studies, we demonstrate the effectiveness of VPBs in terms of performance and computational cost within various learning tasks, namely clustering, classification and change point detection.
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