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Can anybody point me to a survey paper on "Large $p$, Small $n$" results? I am interested in how this problem manifests itself in different research contexts, e.g. regression, classification, Hotelling's test, etc.

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I don't know of a single paper, but I think the current book with the best survey of methods applicable to $p\gg n$ is still Friedman-Hastie-Tibshirani. It is very partial to shrinkage and lasso (I know from a common acquaintance that Vapnik was upset at the first edition of the book), but covers almost all common shrinkage methods and shows their connection to Boosting. Talking of Boosting, the survey of Buhlmann & Hothorn also shows the connection to shrinkage.

My impression is that, while classification and regression can be analyzed using the same theoretical framework, testing for high-dimensional data is different, since it's not used in conjunction with model selection procedures, but rather focuses on family-wise error rates. Not so sure about the best surveys there. Brad Efron has a ton of papers/surveys/book on his page. Read them all and let me know the one I should really read...

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What do you mean by "result" theoretical result ? or numerical results ?

I like the reviews of Jianqing Fan see for example this one and this one on classification (lot of self citations).

Also there are non review paper that make rich review in introduction, see for example this one and this one.

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If you want a resume maybe this is the best you can get

http://www.springer.com/statistics/statistical+theory+and+methods/book/978-3-642-20191-2

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Chapter 18 of Hastie, Tibshirani, and Friedman (12th printing/second edition, this chapter wasn't in the first edition) is a nice overview with some interesting data sets. It's not quite as thorough as their treatment of older material, and a lot of the time they have to give somewhat heuristic explanations of why certain algorithms work better than others. I found it very useful coupled with reading papers for things you want to know more in-depth.

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