When persistence diagrams are formalized as the Mobius inversion of the birth-death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications. The direct definition in the 2-parameter setting, and the corresponding brute-force algorithm to compute them, require $\Omega(n^4)$ operations. But the size of the generalized persistence diagram, $C$, can be as low as linear (and as high as cubic). We elucidate a connection between the 2-parameter and the ordinary 1-parameter settings, which allows us to design an output-sensitive algorithm, whose running time is in $O(n^3 + Cn)$.

翻译：当固态图像莫比乌斯对生死功能的倒置那样正式化时,它们自然会概括到多参数设置中,并享有我们在应用中预期的许多关键特性,例如稳定性。 2参数设置中的直接定义以及相应的粗力算法需要$\Omega(n ⁇ 4) 操作。但通用持久性图表的大小($C$)可以像直线(和立方体一样高)一样低。我们阐明了2参数与普通的1参数设置之间的联系,使我们能够设计出一种对产出敏感的算法,其运行时间为$(n ⁇ 3+Cn) 。