Say I have a set of data points, and applied some form of linear dimensionality reduction. I know that this projection is linear, so there must be a projection matrix unknown to me.
Question :
How do I obtain the Projection Plane (anything that represents it. e.g., its normal vector or its bases) to which the original data was projected?
Context :
The reason why I want the bases is because I have additional points I want to project onto the plane afterwards, but can't have those points affect the outcome of the initial dimensionality reduction. Hence the transformation Matrix is fixed but unknown. I also want to give the plane some perturbation.
Maybe this problem can be thought of as: For an unknown $m \times n$ matrix $A$ in $A\textbf{x} = \textbf{y} $, $\textbf{x} \in \mathbb{R}^n, \textbf{y} \in \mathbb{R}^m, n > m$, (but lets say $n=3, m=2$ for simplicity), where many set of slutions $(\textbf{x}_i, \textbf{y}_i)$ are given, find a set of bases(or one normal vector) representing one projection plane that satisfies $A\textbf{x} = \textbf{y} $ (in which case the row vectors of $A$ would be the bases presumably).
I found similar questions essentially asking the same thing:
- 2D plane to determine transformation matrix?
- 3D projection coordinates onto 2D plane to determine transformation matrix?
both of which are generalized versions of my question, but there either hasn't been a clear solution, or was deemed impossible.
I think the problem lied in that the projection plane is not unique (infinite solutions of planes parallel to one another, having the same projection results), thus the transformation matrix $A$ is not unique. However, I don't care which one of the solutions I have, as long as I have one solution. For example, I have no problem adding a constraint like: for $\mathbf{x}=0, A\mathbf{x}=\mathbf{0}$ which would reduce the number of unknown variables (I think).
Can anyone help?
Edit: added context.