论文标题
多参数持久模块的投影距离
Projected distances for multi-parameter persistence modules
论文作者
论文摘要
依靠捆绑理论，我们介绍了预计的条形码和多参数持久模块的投影距离的概念。投影的条形码定义为在$ \ mathbb {r} $上的持久模块的派生推送。预计距离有两种口味：整体捆圆形指标（ISM）和切成薄片的卷积距离（SCD）。我们对投影条形码的稳定性进行了系统的研究，并表明纤维条形码是投影条形码的特定实例。我们证明ISM和SCD为卷积距离提供了下限。此外，我们表明，可以使用专用于单参数持久性模块的TDA软件来计算$γ$ - 线性ISM和$γ$ - 线性SCD。此外，计算这两个指标所需的时间和内存复杂性是有利的，因为我们的方法不需要计算或存储整个$ n $ persistence模块。
Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the $γ$-linear ISM and the $γ$-linear SCD which are projected distances tailored for $γ$-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire $n$-persistence module.
