I wonder if there's an underlying "random attempts" theory that can tie these successes (that have intrigued me for a long time):

1. Shannon's random code book construction (for achieving the noisy channel rate limit)

2. Johnson-Lindenstrauss lemma http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_l..., the result that underlies most if not all of the compressive sensing field (and all of the recent superfast large scale SVD, PCA algorithms)

And of course this recent result might also be related somehow.