We introduce the new notions of projected distances and projected barcodes 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：我们为多参数持久性模块引入了预测距离和预测条形码的新概念。 预测条形码被定义为将持久性模块推向推向$\mathb{R}$。 预计距离分两种口味: 整体沙夫测量值(ISM) 和切片变速距离(SCD) 。 我们对预测条形码的稳定性进行系统研究, 并显示纤维条形码是预测条形码的一个特例。 我们证明, ISM 和 SCD 为聚合距离提供了较低的界限。 此外, 我们显示, 用于为 $\gamma$- 线性IMM 和 $\gamma$- 线性SCD 预测的距离可以使用用于 $\gamma $- sheave 的TDA 软件来计算。 此外, 用于计算这两条形持续性模块所需的时间和记忆复杂性是有利的, 因为我们的方法不需要计算或存储一个完整的 $- $ peristence 模块。