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Detecting and recovering a low-rank signal in a noisy data matrix is afundamental task in data analysis. Typically, this task is addressed byinspecting and manipulating the spectrum of the observed data, e.g.,thresholding the singular values of the data matrix at a certain criticallevel. This approach is well-established in the case of homoskedastic noise,where the noise variance is identical across the entries. However, in numerousapplications, the noise can be heteroskedastic, where the noise characteristicsmay vary considerably across the rows and columns of the data. In thisscenario, the spectral behavior of the noise can differ significantly from thehomoskedastic case, posing various challenges for signal detection andrecovery. To address these challenges, we develop an adaptive normalizationprocedure that equalizes the average noise variance across the rows and columnsof a given data matrix. Our proposed procedure is data-driven and fullyautomatic, supporting a broad range of noise distributions, variance patterns,and signal structures. We establish that in many cases, this procedure enforcesthe standard spectral behavior of homoskedastic noise -- the Marchenko-Pastur(MP) law, allowing for simple and reliable detection of signal components.Furthermore, we demonstrate that our approach can substantially improve signalrecovery in heteroskedastic settings by manipulating the spectrum afternormalization. Lastly, we apply our method to single-cell RNA sequencing andspatial transcriptomics data, showcasing accurate fits to the MP law afternormalization. Our approach relies on recent results in random matrix theory,which describe the resolvent of the noise via the so-called Dyson equation. Byleveraging this relation, we can accurately infer the noise level in each rowand each column directly from the resolvent of the data.

The Dyson Equalizer: Adaptive Noise Stabilization for Low-Rank Signal Detection and Recovery