Entropic CLT and phase transition in high-dimensional Wishart matrices

@article{Bubeck2015EntropicCA,
  title={Entropic CLT and phase transition in high-dimensional Wishart matrices},
  author={S{\'e}bastien Bubeck and Shirshendu Ganguly},
  journal={ArXiv},
  year={2015},
  volume={abs/1509.03258}
}
We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the… 
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