this is my first post on this exchange so please be nice :)
Here goes: I applied a dimensionality reduction technique called "Linear Discriminant Analysis" (LDA) on a dataset to reduce it's high dimensionality. In this subspace I want to be able to find which class vectors are the most closely related (the hardest to differentiate). For this purpose I need to find the minimum distance between all the distances between these class vectors.
So my question boils down to: "how do I compute distances between vectors in a high dimensional case?"
EDIT: The context is facial recognition. I am using a type of LDA by Fisher to extract features from images (reducing the dimensionality and maximizing the distances between different classes, while minimizing the within class-scatter) and then I apply Classification with a simple kNN approach (looking at the neighbours in the subspace)
EDIT2: After performing Fisher's LDA, I have c class vectors (c here is equal to the number of persons in my dataset). I want to perform classification in this space. I want to see which classes are closely related because these classes (let's say faces) will be hard to differentiate between and accurate classification between these two faces will be hard.
If
$$ {\bf x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} $$
with a similar expression for ${\bf y}$, then you can define the distance as
$$ |{\bf x} - {\bf y}|_2 = \left[\sum_{k=1}^n(x_k - y_k)^2\right]^{1/2} $$
This is a 2-norm (hence the subscript), you can use numerically cheaper distances, such as the 1-norm
$$ |{\bf x} - {\bf y}|_1 = \sum_{k=1}^n|x_k - y_k| $$