Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
   \{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let us be given an finite family of non-injective matrices
\begin{align*}
   \{M_j \in \mathbb R^{m \times n} : j = 1, \dots, J\},
\end{align*}
e.g. $m<n$.

In a nutshell, the problem I would like to address is the following problem: For any $j = 1, \dots, J$ we are given the set of points (i.e. no knowledge about ordering!)
\begin{align*}
   \{M_j x_1 , \dots, M_j x_N\}
\end{align*}
which can be seen as a projection of the set $\{x_1, \dots, x_N\}$. 

My question is under which conditions on the family of projections matrices we can uniquely reconstruct the set $\{x_1, \dots, x_N\}$. Intuitively I would say that $J$ has to be large enough (dependend on $N$) and that the matrices should fullfill some assumption like
\begin{align*}
   \bigcap_{j = 1,\dots, J} \ker M_j = \{0\}.
\end{align*}