reference request for a book on high dimensional probability and data analysis written for mathematicians

It has been some time since the question was asked and I believe the recent text "High Dimensional Probability" by Roman Vershynin might be just what you are looking for. It covers a variety of topics, focusing on the concentration phenomenon on higher dimensions. It is both theoretically heavy and intuitionally approachable.

Its current draft is also available for free, in the author's webpage. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html#

A recently published book by Wainwright, on high dimensional data, which seems useful

High-dimensional statistics : a non-asymptotic viewpoint