Before revision:  In this article , we study the differentiability of a general functional associated with the most common topological construction, that is, the persistence map , and we prove a convergence result of stochastic subgradient descent for such a functional .
After revision:  In this article , we study the differentiability of a general functional associated with the most common topological construction, that is, the persistence map , and we prove a convergence result of stochastic subgradient methods for such functions .