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Let's say I'm trying to cluster $n$ points in $\mathbb{R}^p$, and I know in advance that only $s$ many of these $p$ dimensions determine the differences between the clusters. Of course, I don't know which of these dimensions are the important ones.

All in all, I have an unsupervised learning problem (clustering) and I'm trying to do variable selection while also trying to determine the clusters. Are there any well known algorithms for this? Or better, any theoretical results?

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I'm not aware of any clustering algorithm which does global feature selection. Usually, you would do this in preprocessing.

There are the closely connected domains of subspace clustering and correlation clustering and biclustering, all of which consider that different parts of the data set may need different features (and not need the other features).

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The idea of simultaneous feature selection and clustering was first explored in psychometrics in the 80's. Two algorithms that come to my mind are SYNCLUS and my own GROUPALS. If you search for references to these papers, then it is easy to form a picture of the current state of the art.

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