  A recent line of work has studied the relationship between the Wishart matrix
$X^\top X$, where $X\in \mathbb{R}^{d\times n}$ has i.i.d. standard Gaussian
entries, and the corresponding Gaussian matrix with independent entries above
the diagonal. Jiang and Li (2015) and Bubeck et al. (2016) showed that these
two matrix ensembles converge in total variation whenever $d/n^3\to \infty$,
and Bubeck et al. (2016) showed this to be sharp. In this paper we aim to
identify the precise threshold for $d$ in terms of $n$ for subsets of Wishart
matrices to converge in total variation to independent Gaussians. It turns out
that the combinatorial structure of the revealed entries, viewed as the
adjacency matrix of a graph $G$, characterizes the distance from fully
independent. Specifically, we show that the threshold for $d$ depends on the
number of various small subgraphs in $G$. So, even when the number of revealed
entries is fixed, the threshold can vary wildly depending on their
configuration. Convergence of masked Wishart to independent Gaussians thus
inherently involves an interplay between both probabilistic and combinatorial
phenomena. Our results determine the sharp threshold for a large family of $G$,
including Erd\H{o}s-R\'enyi $G\sim \mathcal{G}(n,p)$ at all values $p\gtrsim
n^{-2}\mathrm{polylog}(n)$. Our proof techniques are both combinatorial and
information theoretic, which together allow us to carefully unravel the
dependencies in the masked Wishart ensemble.
