De Finetti-Style Results for Wishart Matrices: Combinatorial Structure and Phase Transitions

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.