Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.

My question is, given $(X,d)$ and some $\epsilon\in(0,1)$, under what conditions can we deduce that there exists some other metric space $(Z,\rho)$ such that $(X,d)$ is the $\epsilon$-snowflake of $(Z,\rho)$; i.e.: $$ (X,d)=(Z,\rho^{\epsilon})? $$