  We introduce a new method for obtaining quantitative convergence rates for
the central limit theorem (CLT) in a high dimensional setting. Using our
method, we obtain several new bounds for convergence in transportation distance
and entropy, and in particular: (a) We improve the best known bound, obtained
by the third named author, for convergence in quadratic Wasserstein
transportation distance for bounded random vectors; (b) We derive the first
non-asymptotic convergence rate for the entropic CLT in arbitrary dimension,
for general log-concave random vectors; (c) We give an improved bound for
convergence in transportation distance under a log-concavity assumption and
improvements for both metrics under the assumption of strong log-concavity. Our
method is based on martingale embeddings and specifically on the Skorokhod
embedding constructed by the first named author.
