I want a projection from nD to 2D, such that all the identity basis vectors are equally spaced. My initial idea was to use PCA on the identity basis. This seems to work well for 3D
Specific to 3D I think this is an axonometric/isometric projection:
But surely we can generalize this type of projection to higher dimensions? Here's the simple solution.
- Basis Vector 1: $q_i = cos(2i\pi/n/\sqrt{n/2}), \vec q\in \mathbb R^n$
- Basis Vector 2: $p_i = sin(2i\pi/n/\sqrt{n/2}), \vec p\in \mathbb R^n$
We note this satisfies $\vec{1} \cdot \vec q = 0$, $\vec{1} \cdot \vec p = 0$, $\vec p \cdot \vec q = 0$. This does not match PCA for higher dimensions.
So my question is 2-fold.
- This is probably not a unique projection, what's the typical name for this? If there is no name for it, what's the best way to describe this projection? I can't seem to find one googling around.
- (and bonus) Why doesn't PCA match this projection scheme? I know PCA maximizes variance, and the total spread seems larger in the PCA case, but there are a bunch of clustered points, what's up with that? Is my understand of variance misplaced? What are we maximizing in our direct solution if not variance?