EDIT: I'd like to reformulate my primary question.
I have a set of points in $\mathbb{R}^D$ and I reduce the dimensions of the points to some $\mathbb{R}^L$. I do this with multiple configurations where I set the lower dimension size L to different values.
Say I lower the dimensions of the points p and q to $\mathbb{R}^2$ and then to $\mathbb{R}^3$ and receive $p^{(2)}, q^{(2)}$ and $p^{(3)}, q^{(3)}$. Now I calculate the euclidean distance of $p^{(d)},q^{(d)} \: \: \forall d \in \{2,3\}$ to get $d^{(2)}$ and $d^{(3)}$.
Are these distances comparable, or will $d^{(3)}$ generally be bigger than $d^{(2)}$?
Background:
I am using a supervised model to cluster data points. The goal is to have tight clusters. I would like to compare different models, but the problem is that they output the data points in different number of dimensions, like illustrated:
My metric to minimize is the sum of the distances between the points and the centroid of the cluster:
$$ \sum_{i}^{N}\sqrt{\sum_{j}^{D}(x_{ij} - centroid_j)^2} $$
D is the number of dimensions and N the number of data points.
So my question is, how do I know which model is doing a better job in clustering the points? I am pretty sure that I can't simply compare the metric mentioned above. How could I achieve this?