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续模的投影条码和投影距离的概念。投影条码定义为持续模到$\mathbb{R}$上的导出前推。投影距离有两种形式：积分层度量（ISM）和切片卷积距离（SCD）。我们系统研究了投影条码的稳定性，并证明了纤维条码是投影条码的一个特例。我们证明了ISM和SCD是卷积距离的下界。此外，我们表明针对$\gamma-层的投影距离——即$\gamma$-线性ISM和$\gamma$-线性SCD——可以使用专用于单参数持续模的TDA软件进行计算。而且，计算这两种度量的时间和内存复杂度具有优势，因为我们的方法无需计算或存储整个$n$参数持续模。