Blind Noise Equalization for Large Data Matrices under General Variance Profiles

Boris Landa, Francesco Strino, Fabio Parisi, Ruiqi Li, Yuval Kluger

Low-rank approximation is a basic tool for inference from large and noisy data matrices. However, when noise variances differ across entries, a setting known as heteroskedasticity, the noise can distort the observed spectrum, producing spurious components and obscuring weak signals. To address these challenges, we develop BRINE (Blind Resolvent-based Iterative Noise Equalizer), a data-driven procedure that iteratively rescales rows and columns to equalize the average noise variance across both dimensions. Under suitable conditions, this procedure restores the Marchenko-Pastur law for the noise spectrum and can amplify signal components without prior knowledge of the signal or noise structure. A key idea underlying our approach is to equalize the diagonal entries of the data resolvent as a proxy for equalizing the unknown noise variances. At the population level, we identify a potential function that characterizes this equalization and motivates an iterative minimization procedure. We prove convergence of the iterations to the true equalizer for arbitrary positive variance profiles, with a geometric rate under mild conditions. The updates can be estimated directly from noisy data,…