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

TL;DR

This paper introduces an adaptive, data-driven normalization method based on random matrix theory to stabilize heteroskedastic noise, enabling improved detection and recovery of low-rank signals in noisy data matrices.

Contribution

The paper develops a novel normalization procedure using the Dyson equation to equalize noise variance, enhancing spectral analysis for heteroskedastic noise in data matrices.

Findings

Normalization enforces the Marchenko-Pastur law in heteroskedastic noise.

Improved signal detection and recovery after normalization.

Successful application to biological data like single-cell RNA sequencing.

Abstract

Detecting and recovering a low-rank signal in a noisy data matrix is a fundamental task in data analysis. Typically, this task is addressed by inspecting and manipulating the spectrum of the observed data, e.g., thresholding the singular values of the data matrix at a certain critical level. This approach is well-established in the case of homoskedastic noise, where the noise variance is identical across the entries. However, in numerous applications, the noise can be heteroskedastic, where the noise characteristics may vary considerably across the rows and columns of the data. In this scenario, the spectral behavior of the noise can differ significantly from the homoskedastic case, posing various challenges for signal detection and recovery. To address these challenges, we develop an adaptive normalization procedure that equalizes the average noise variance across the rows and columns of a given data matrix. Our proposed procedure is data-driven and fully automatic, supporting a broad range of noise distributions, variance patterns, and signal structures. Our approach relies on recent results in random matrix theory, which describe the resolvent of the noise via the so-called Dyson equation. By leveraging this relation, we can accurately infer the noise level in each row and each column directly from the resolvent of the data. We establish that in many cases, our normalization enforces the 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 signal recovery in heteroskedastic settings by manipulating the spectrum after normalization. Lastly, we apply our method to single-cell RNA sequencing and spatial transcriptomics data, showcasing accurate fits to the MP law after normalization.