I have a metric space $(X,d)$. I have a physical situation (data) where each physical entity corresponds to an $x \in X$.
I want to do some mathematical/statistical modeling of this data, but the problem is I cant add two elements of this set, as addition is not defined on them or its not closed under addition operation. So i take a strange approach, where I cluster the data, $X$ (using the metric $d$) into $N$ clusters, each cluster $C_i$ having a centroid $K_i$. Now I give a vector space like representation to each element $x \in X$ as the vector $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)] \in V$$ i.e, $x$ is represented by a set of distances from each of the $N$ centroids. This way we moved to a vector space from a metric space, there by enabling us to do some modeling in vector space. After modeling when we get a final new vector $p$, it may not be having any corresponding element in our data $D$, thus we assume such $x$ that $||x_v -p||$ is minimum over entire $D$ or some selected codebook (subset of $D$).
What want is some form of mathematicaly rigourous formulation of this problem in a formal way, if possible making any suitable assumptions.