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Let's say that I have a set of datapoints. Let's say I also have a set of pairs of these points, where a single point can be in multiple pairs. Let's say I also have a set of triples, etc with the same rules. What do I call each of these? I don't think I can identify them as clusters because a single point can be in multiple pairs(or triple etc) with points that are not themselves related by such a pair(or triple etc).

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Outside of machine learning, the terminologi is graphs and hypergraphs. In some articles in machine learning, I have seen the word "Type" used as a general kind of relation, but this is probably not a correct terminology. –  utdiscant Apr 21 '12 at 22:36
    
Yeah, graph is too "mathematical". I know that they can be described as that but I'm looking for a more statistical word. –  Andrew Apr 21 '12 at 22:41
    
Thanks anyways! –  Andrew Apr 21 '12 at 22:41
    
In the applied machine learning literature, 'hypergraph' is the appropriate term for most applications, and it is meant in the full mathematical sense. You see that in everything from stats papers to computer vision to social networks. See my answer below. –  EMS Apr 21 '12 at 22:44
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This is an unusual question. To my knowledge, it depends on the context you're speaking in. If you model your data points as nodes in a graph, then the things you're talking about are merely edges in a hypergraph over those nodes. And this is frequently used in machine learning.

For example, in event detection in video analysis, the primitives might be descriptors of video segments. Say your video is depicting cars coming and going from a parking lot. Then it makes sense to have primitives for items like 'car_idling' or 'car_entering' or 'car_exiting'. But more complex actions could be created by looking at ordered (or unordered) collections of these, such as {'car_entering', 'car_idling', 'car_exiting'} = 'car_dropoff', or something like that.

I've seen a paper doing exactly the above analysis on car/parking lot surveillance camera data. I can't see to find that specific reference, but here is a PDF of another vision paper that uses the hypergraph approach to action detection.

When folks build large Hidden Markov Models on top of these primitives, it can be shown that the analysis is fully equivalent to learning optimal weights for edges in an associated hypergraph. So in that setting, that would be the appropriate language for your problem.

But in other settings it can be different. Simplest of all, if your data points were just real scalars, you could simply be looking for combinations drawn from a sample set. It really depends on what your context it.

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