Research in high-dimensional statistics vs. machine learning?

Question

(I've posted this question on CV, but I feel it would also be great to hear from experts in DS community.)

As a PhD student starting to think about dissertation topics, I am particularly interested in high-dimensional statistical learning. I wish to find some research review/survey/papers (or webpages, blogs, whatever...) about the state-of-the-art research in this research area, but there seems limited resources I can obtain. My first question is then,

1. Could you describe some current interesting research topics in high-dimensional statistics? If you can list any relevant resources (papers, webpages, etc.), that would be really helpful.

In addition, I've noticed that high-dimensional statistical learning is closed related with machine learning research. For example, the idea of penalized regularization in high-dimensional statistics was used in machine learning domain, like support vector machine, boosting tree, (sparse) additive models, etc. My question is,

2. what are good research papers about the interplay of high-dimensional statistics and machine learning?

Last, since high-dimensional statistics was really motivated by genetic research (like gene-expression analysis, or genome-wide association studies), most of applications in high-dimensional research are devoted in that area.

3. Are there any successful applications of high-dimensional statistics in areas other than genetics, particularly say, image/text mining, recommendation, etc, areas where machine learning techniques have long been used?

4. A new question to machine learning researchers/practitioners: I might be wrong, but as I understand, most machine learning algorithms are designed for low-dimensional problems (or at least the number of features is smaller that number of observations). Are there any successful applications of machine learning techniques for modeling high-dimensional data?

Any resources/comments are highly appreciated. Thanks.

Answer 1

This makes me think right off the bat of Gaussian processes.

The GP was initially developed by geostatisticians in the 1970s as a model for the distribution of resources over a geographic area; this process is known as kriging (or also Gaussian process regression). But more recently GPs have been used pretty widely in the movement/control/robotics area to model movements and trajectories of robotic limbs etc. over time.

Statistically, GPs are treated as priors over vectors of infinite dimension, such that the distribution of any subset of the values of a vector is Gaussian distributed. There's an online book if you want to read more!