Denoising clustering covariance matrices with Rotational Invariant Estimators

Abstract

Cosmological parameter inference from galaxy clustering relies critically on accurate estimates of the covariance and precision matrices. These are often obtained from a limited number of mock catalogs, introducing noise and bias in the precision matrix when the data-vector dimension becomes comparable to the number of available realizations. We present the first application of the Rotational Invariant Estimator (RIE) to the large-scale clustering of galaxies, benchmarking it against the standard sample covariance and the non-linear shrinkage estimator NERCOME for both the two-point correlation function (2PCF) and power spectrum. Using controlled synthetic data sets with analytically known covariance matrices, we estimate the covariance with all three methods across a range of mock-to-dimension ratios $q = N/D$ and data-vector sizes $D$. We then perform Bayesian inference with an EFT-based model and quantify each estimator through the Figure of Bias (FoB) and Figure of Merit (FoM). After correction for finite-$N$ effects, the sample covariance recovers unbiased average uncertainty volumes but suffers from growing best-fit scatter and bias at small $q$ due to the Dodelson--Schneider effect. Both NERCOME and RIE substantially reduce these stochastic shifts; however, the uncertainties they assign are probe-dependent. In configuration space, both estimators can yield overly tight constraints, with a bias that grows with $D$. In Fourier space, RIE delivers markedly improved best-fit stability with only mild FoM bias, whereas NERCOME tends to overestimate the constraining power. Among the estimators tested, RIE emerges as the most effective at stabilizing best-fit recovery, particularly in Fourier space, where it closely reproduces the reference posteriors even when the number of mocks barely exceeds the data-vector dimension.