  We propose convex optimization algorithms to recover a good approximation of
a point measure $\mu$ on the unit sphere $S\subseteq \mathbb{R}^n$ from its
moments with respect to a set of real-valued functions $f_1,\dots, f_m$. Given
a finite subset $C\subseteq S$ the algorithm produces a measure $\mu^*$
supported on $C$ and we prove that $\mu^*$ is a good approximation to $\mu$
whenever the functions $f_1,\dots, f_m$ are a sufficiently large random sample
of independent Kostlan-Shub-Smale polynomials. More specifically, we give
sufficient conditions for the validity of the equality $\mu=\mu^*$ when $\mu$
is supported on $C$ and prove that $\mu^*$ is close to the best approximation
to $\mu$ supported on $C$ provided that all points in the support of $\mu$ are
close to $C$.
