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 . Building on real analytic geometry arguments, we propose a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way. We also provide a simple, explicit and sufficient condition for convergence of stochastic subgradient descent for such a functional .