I want to visualize high dimensional vectors (n~=10-200) using 2 dimension in a diagram. Over time more and more vectors are added and similar vectors should build clusters, so that it is possible to see changes over time.
Therefore I am looking for an LP-norm ($f$) which maps a vector to a tuple like this:
$$f:\mathbb{R}^n \to \mathbb{R} × \mathbb{R}$$
Using variables for the norm like f(x)=(y,z) the trivial case that $∀ x: y=z$ should be ruled out. Another restriction is that the derivative $\frac{\partial y}{\partial x}$ and $\frac{∂ z}{∂ x}$ should be continuous. In other words: changing x a little bit should not result in jumps when plotting y and z in a diagram.
Does a concept exist like this?
What I already thought of:
I know that a principal component analysis can be used for this, but a PCA can only be performed after I got enough points.
I tried splitting a vector in two halves and calculated a norm for each half but it seems that I get some strange curves and jumps and this maybe breaks the definition of a norm.
It seems that I was wrong that using two norms for each half of the vector does not give proper results. The solution is rather trivial. For some "proof" I wrote a script to do a random walk in high dimensional space and used the 1-norm for each half of the vector. The jumps I saw in my original application must have had another cause.