# Finding a Projection Plane in Dimensionality Reduction (e.g., Multidimensional Scaling)

I have a set of data points in high-dimensional space that I wish to map onto a lower dimension (3D or 2D).

Question :

How do I obtain the Projection (Hyper)Plane (e.g., its normal vector or its set of bases) on which the data was mapped?

Context :

As a part of an experiment I'm conducting, I am comparing different techniques of dimensionality reduction (e.g., PCA, MDS, ICA) on a same given set of data. After mapping to lower-D, I want to add some perturbations to the projection plane and re-project the same data to the perturbated plane. In order to do so, I need the normal vector(or bases) representing the projection plane.

It's easy for techniques like PCA because the principal components are what I want, but trickier for stuff like MDS that doesn't use a direct projection as its approach.

This question asks the same as this past question, but with a different motive.

Also, mathematically speaking, I assume there would be an infinite number of solutions to this problem, since the Projection Plane representing the transformation by MDS (or other techniques) would not be a unique plane(I think there would be an infinite number of planes parallel to one another as a solution), because it is an under-constrained problem. But I'm fine with having just one solution (one projection plane I can use for perturbation)