Given a collection of points in 3d. The value of each dimension can be visualized
When we visualize these points in 3d. They clearly are actually only live on a plane (but not perfectly), visualized bellow
My question is, what is a general approach to verify that any number of collection of points live on the same plane. Or more generally, any collection of points of dimenion m, actually live on n dimension space where n < m.
One way of doing such a job is to find 3 points to make a plane and iteratively test every point. However, this is not what I am looking for due to the fact that it is not generalizable for more than 3->2 dimension and if there is a slight amount of noise, the method falls apart. One approach I have tried is to find the SVD decomposition and decide by the eigenvalues. I am certain this problem has been solved principally but I am not sure how, I appreciate any opinion.
Total Least Squares technique is what you need. If the points are almost layered into one plane (in any dimension), that method will reveal it and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.