Find parametric representation of cluster surface [closed]

I have a large set of records with each record containing $N$ variables $v_i$ (say $N=20$).

I can transform these variables $v_i$ to be in the range [0,1] and am able to use a simple loss function $L = \sum_i v_i^2$. After that I know that there are two large clusters of points with small $L$ and large $L$. There are also few points in between these clusters. I can successfully assign records to these clusters.

But what R functionality would I use to get a parametric representation (in the variables $v_i$) of the surface of the "large $L$ cluster", preferably as a function of $L$?

(I know I could train a neural network on my data to give me a similar answer, but I would like to stay away from that for this application.)

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(1) Are these clusters in L alone or clusters in N -dimensional space? (2) The penultimate remark about "a function of L " may be confusing, because it seems to conflict with what a "parametric representation ... in ...v[i]" would be (namely, a smooth map from an M << N dimensional space into N dimensional space whose image closely approximates most of the records). – whuber Aug 18 '10 at 18:53
@whuber: (1) The points are in both in L and the N-dimensional space of the v_i. However since I have more than a few variables most points lie near the surface and using simple box cuts e.g. v_i>0.5 would miss a lot of points that would have a "good enough" L. (2) Think of L as the onion layer I would be sitting on. – honk Aug 18 '10 at 19:47
I'm still confused by the question, because it uses some words in ways that are inconsistent with the definitions I know. A "parametric representation" of a (hyper)surface in $N$ dimensions is a function of $N-1$ variables, for instance. A "function of $L$" couldn't do anything for you in this regard. I'm thinking that perhaps you're looking for $L$ itself to be a function of $N-1$ angular parameters. Could you perhaps clarify this or give an example of what you're looking for as a solution? – whuber Sep 16 '10 at 3:12
@Honk answers to whuber's questions are crucial to being able to answer this question, I notice you regularly pop by after a few months of absence and were even a beta participant, so I'll leave this for a while, but if you are not able to improve the question, I think we have to close this question as unanswerable. – Corone Feb 17 at 19:49
@Corone: Thanks for wrangling old questions. I tried to answer whuber's questions earlier and wasn't able to add to what I had said. However, this question is specifically tied to how R internally parametrizes clusters, and given the many different algorithms giving a generally correct answer might be impossible. If you think this question should be closed and have no issue with that. – honk Feb 17 at 21:03

closed as not a real question by gung, Macro, whuber♦Feb 18 at 17:59

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