Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes

Abstract

The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp \{-\frac{1}{4}{n}^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt{n}$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\mathrm{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. Distinct off-diagonal entries are exactly pairwise independent under every $\mathrm{LKJ}(\eta )$ law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.

arXiv comment45 pages, 5 figures, 3 tables (including supplement). v2: new subtitle; Chen-Stein point-process limit extended to the whole LKJ family; stronger nearest-correlation repair result (Theorem 5); revised exposition