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*}