Well, lets try. For the embeddings, you'll have to search the internet for details, and this may be not a "proper" ML. But the general idea is that if you have some nice euclidean algorithm, like k-means, and you want to cluster some data w.r.t a different metric, you could try putting the data in the euclidean space without distorting the distances too much and then apply the algo. there. For k-means in particular the nearest neighbor search in Euclidean space can be made efficient, I think. This didn't take off much because most of the results were unfortunately negative. Although there were some recent positive results, even with discussion here [1]. Bourgain had some early results on the embeddability of finite metric spaces into classical spaces.

The concentration of covariance matrices is fundamental for most "spectral" methods, in particular spectral clustering. See for instance [2]. People proved and reproved these concentration results a lot of times, but Bourgain's was the earliest with N\log N sample complexity for log concave measures, improving on a result of Lovazs et al. (!!!) More importantly, the method with which he proved this, controlling the concentration by decomposing into bounded and "rarely non-zero" parts is now all around in statistical learning theory. So each time people want to prove uniform law of large numbers and nothing standard works, people try to find a good decomposition.

[1] https://news.ycombinator.com/item?id=17752836 [2] Clustering with Spectral Norm and the k-means Algorithm, Amit Kumar and Ravindran Kannan (Section 6...)