Formalism for moving from a metric space into a vector space for mathematical/statistical modeling given a data

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.

• Its a bounded metric space. – Rajesh Dachiraju Jan 18 '14 at 6:31
• How can I represent my finite data points in the metric space and also clustering, Can i consider some sort of molecules, like they are considered for Arens-Eells space. Appreciate your help in this regard. – Rajesh Dachiraju Jan 19 '14 at 6:35

Your idea reminds me of a standard construction which is often used to prove the existence of a completion of an arbitrary metric space. If a metric space $X$ is bounded, you can embed it into the Banach space $C_b(X;R)$ of bounded continuous functions from $X$ to $R$: simply identify $x\in X$ with the function $f_x(y)=d(x,y)$. If $X$ is not bounded it is easy to modify this construction: fix a reference point $x_0$ and define $f_x(y)=d(x,y)-d(x_0,y)$.
• Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and these points are selected by clustering. I'd like some help in putting this approximation in rigorous mathematical terms, to see if I am missing anything crucial. – Rajesh Dachiraju Jan 19 '14 at 6:49
• moreover I am interested in $||.||_2$ norm than supremum norm. – Rajesh Dachiraju Jan 19 '14 at 12:27