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 $\gamma$-linear ISM and the $\gamma$-linear SCD which are projected distances tailored for $\gamma$-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.

翻译：基于层论，我们引入了多参数持久性模的投影条码与投影距离概念。投影条码定义为持久性模在$\mathbb{R}$上的导出推前。投影距离分为两类：积分层度量（ISM）与切片卷积距离（SCD）。我们对投影条码的稳定性进行了系统研究，并证明纤维条码是投影条码的一个特例。我们证明了ISM和SCD为卷积距离提供了下界。此外，我们证明专为$\gamma$-层设计的投影距离——$\gamma$-线性ISM与$\gamma$-线性SCD——可利用专用于单参数持久性模的TDA软件进行计算。同时，计算这两种度量的时间与内存复杂度具有优势，因为我们的方法无需计算或存储整个$n$参数持久性模。