  We introduce and systematically study a profile function whose asymptotic
behavior quantifies the dimension or the size of a metric approximation of a
finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_{\alpha},
d_{\alpha}, k_{\alpha}, \varepsilon_{\alpha })\}_{\alpha\in I, }$ where each
group $G_\alpha$ is equipped with a bi-invariant metric $d_{\alpha}$ and a
dimension $k_{\alpha}$, for strictly positive real numbers $\varepsilon_{\alpha
}$ such that $\inf_{\alpha }\varepsilon_{\alpha }>0$. Through the notion of a
residually amenable profile that we introduce, our approach generalizes
classical isoperimetric (aka Folner) profiles of amenable groups and recently
introduced functions quantifying residually finite groups. Our viewpoint is
much more general and covers hyperlinear and sofic approximations as well as
many other metric approximations such as weakly sofic, weakly hyperlinear, and
linear sofic approximations.
