A multi-eigenbasis approach to covariance matrix denoising for cosmological inference
Accurate covariance matrix estimation is crucial to cosmological analyses, enabling unbiased parameter inference with well-calibrated uncertainties. Obtaining a reliable estimate generally requires far more independent samples than the dimension of the data vector, which is not always feasible. This challenge is especially relevant for the 3D Ly$\alpha$ forest analysis, which measures the Ly$\alpha$ auto-correlation and its cross-correlation with quasars in bins of comoving separation to jointly constrain cosmological parameters. The consequence is a very large data vector, and the data-driven covariance measured from sub-samples is non-invertible. The current approach applies a smoothing procedure to the off-diagonals of the correlation matrix to establish invertibility, but this does not fully capture the true correlation structure. In this work, I present a novel multi-eigenbasis denoising method for the data-driven covariance matrix, developed in the context of the 3D Ly$\alpha$ forest analysis and conditioned on DESI DR1 mock simulations. The measured noisy covariance is first projected onto the eigenbasis of a mock-based reference, yielding an initial denoised estimate. A weighted residual correction is then constructed by projecting the noisy residual onto a second eigenbasis derived from a mock-trained classifier, capturing correlation structure not recovered in the initial reconstruction. I validate the method on mock covariances withheld from classifier training and find significant improvements over the current smoothing-based approach in matrix-level reconstruction metrics and in the recovery of cosmological parameter posteriors when compared to those obtained from the true covariance measured from many mock realizations.