Finding a plane numerically Suppose I have three large finite sets $\{x_i\}$, $\{y_i\}$ and $\{z_i\}$;
they are obtained by measuring coordinates of a collection of vectors in $\mathbb{R}^3$, but I do not know which triples correspond to one vector.
Assume I know that all of the vectors lie in a plane, but I do not know which one.
I want to reconstruct this plane and the original collection of vectors.
Evidently the problem can be solved and the solution is unique if the original data is generic.
But I do not see a reasonable algorithm to solve this problem.
(For sure checking all subdivisions into triples is not a good idea.)

Is there a numerical method to find this plane?

and

Did anyone considered this or similar problem; does it have a name?

 A: Here is a reasonable heuristic which you might turn into an algorithm. The idea is to look at generic possibilities inside the bounding box which covers the range of points. We start with imagining the product of three intervals. The first interval is [min x, max x], and we have similar for y and z.
Suppose we end up with a flat box (the z interval is relatively small). Then we know the normal to the plane is close to the z axis, and we can use that information to test various triples and rule out others.
For example, we can try a hexagon test, for example assume the normal is like (-c,-c,1). Then large values of x and y do not belong in the same tuple, as do small values. One can try (x,y,max z) as a test point for x+y close to Max x + mean y or mean y + max x.
Or we can try a ski slope test, where the normal is close to (0,-c,1), in which case x and z are not highly correlated but y correlates with z strongly. The point is that testing for such correlations is done pretty quickly, and can confirm or rule out a ski slope case.
In short, the plane is likely to resemble one case (two of the three sets are highly correlated), or the other (can't see the correlation, so can't have say all three coordinates be Max without saying something strong about the plane) which has implications for what pairs of numbers are allowed in a triple.
Gerhard  "Pretends There Aren't Many Planes" Paseman, 2019.05.20.
A: defining as the initial set of grid points $\lbrace (x_i,y_j,z_k)\rbrace$, where the indices resemble initally given point numbers,
I would suggest to do the following:  
repeatedly


*

*calculate the (geometric) least-squares plane through the remaining grid points  

*remove a certain fraction of the remaining grid points, that are farthest away from the calculated plane  


until all points are on the calculated plane.
