Biwhitening Reveals the Rank of a Count Matrix
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Abstract. Estimating the rank of a corrupted data matrix is an important task in data analysis, most notably for choosing the number of components in principal component analysis. Significant progress on this task was achieved using random matrix theory by characterizing the spectral properties of large noise matrices. However, utilizing such tools is not straightforward when the data matrix consists of count random variables, e.g., a Poisson matrix, in which case the noise can be heteroskedastic with an unknown variance in each entry. In this work, we focus on a Poisson random matrix with independent entries and propose a simple procedure, termed biwhitening, for estimating the rank of the underlying signal matrix (i.e., the Poisson parameter matrix) without any prior knowledge. Our approach is based on the key observation that one can scale the rows and columns of the data matrix simultaneously so that the spectrum of the corresponding noise agrees with the standard Marchenko–Pastur (MP) law, justifying the use of the MP upper edge as a threshold for rank selection. Importantly, the required scaling factors can be estimated directly from the observations by solving a matrix scaling problem via the Sinkhorn–Knopp algorithm. Aside from the Poisson, our approach is extended to families of distributions that satisfy a quadratic relation between the mean and the variance, such as the generalized Poisson, binomial, negative binomial, gamma, and many others. This quadratic relation can also account for missing entries in the data. We conduct numerical experiments that corroborate our theoretical findings, and showcase the advantage of our approach for rank estimation in challenging regimes. Furthermore, we demonstrate the favorable performance of our approach on several real datasets of single-cell RNA sequencing, high-throughput chromosome conformation capture, and document topic modeling.
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Biwhitening is a recently developed method to remove independent heteroskedastic noise in data with inspirations from random matrix theory.
“Biwhitening57 is a recently developed method to remove independent heteroskedastic noise in data with inspirations from random matrix theory.”
From Causal identification of single-cell experimental perturbation effects with CINEMA-OT (2023) · cites this paper as [57]
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Choulakian, V., & Allard, J. (2025) · Analytics · doi:10.3390/analytics4040032
CellMentor: cell-type aware dimensionality reduction for single-cell RNA-sequencing data
Hevdeli, O., Petrenko, E., & Aran, D. (2025) · Nature Communications · doi:10.1038/s41467-025-67088-7
Random Matrix Theory-guided sparse PCA for single-cell RNA-seq dataPreprint
Chardès, V. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.02.26.708218
Random Matrix Theory-Guided Sparse Principal Component Analysis for Single-Cell RNA-Seq Data
Chardès, V. (2026) · Journal of Computational Biology · doi:10.1177/15578666261493157
- Authors
- Landa, B., Zhang, T. T. C. K., & Kluger, Y.
- Affiliations
- Yale University (US)
- University of Pennsylvania (US)
- Venue
- SIAM Journal on Mathematics of Data Science
- Published
- 2022-12-20
- Volume
- 4
- Issue
- 4
- Pages
- 1420-1446
- ISSN
- 2577-0187 (ISSN-L)
- PMC
- PMC10417917
- OpenAlex
- W3139046810
- Cited by
- 20
- References
- 94
- Citations by year
- 2026: 3 · 2025: 7 · 2024: 3 · 2023: 4 · 2021: 3
- FWCI
- 2.58
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- SIAM Journal on Mathematics of Data Science (publishedVersion)
- PubMed (publishedVersion)
- PubMed Central (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Institutes of Health (R01GM131642, UM1DA051410, U01DA053628, U54AG076043, R01GM135928, 2P50CA1219)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Physical Sciences › Engineering › Computational Mechanics › Sparse and Compressive Sensing Techniques
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- rank estimation, Marchenko-Pastur law, heteroskedastic noise, matrix scaling, Sinkhorn-Knopp algorithm, principal component analysis, missing data, single-cell RNA sequencing, chromosome conformation capture, topic modeling
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