When dealing with covariance matrices, one always wonder how thresholding coefficients will yield a matrix with similar properties. Today, we have some answers in that area:

We study the spectra ofp×p random matricesK with off-diagonal(i,j) entry equal ton−1/2k(XTiXj/n1/2) , whereXi 's are the rows of ap×n matrix with i.i.d. entries andk is a scalar function. It is known that under mild conditions, asn andp increase proportionally, the empirical spectral measure ofK converges to a deterministic limitμ . We prove that ifk is a polynomial and the distribution of entries ofXi is symmetric and satisfies a general moment bound, thenK is the sum of two components, the first with spectral norm converging to∥μ∥ (the maximum absolute value of the support ofμ ) and the second a perturbation of rank at most two. In certain cases, including whenk is an odd polynomial function, the perturbation is 0 and the spectral norm∥K∥ converges to∥μ∥ . If the entries ofXi are Gaussian, we also prove that∥K∥ converges to∥μ∥ for a large class of odd non-polynomial functionsk . In general, the perturbation may contribute spike eigenvalues toK outside of its limiting support, and we conjecture that they have deterministic limiting locations as predicted by a deformed GUE model. Our study of such matrices is motivated by the analysis of statistical thresholding procedures to estimate sparse covariance matrices from multivariate data, and our results imply an asymptotic approximation to the spectral norm error of such procedures when the population covariance is the identity.

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