# Empirical Bayes Matrix Factorization

arXiv: https://arxiv.org/abs/1802.06931 · HTML: https://mlsanity.com/paper/1802.06931/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2018.

## Built on by (most cited first, 9 of 9)

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- [MFAI: A Scalable Bayesian Matrix Factorization Approach to Leveraging Auxiliary Information](https://mlsanity.com/paper/2303.02566/) (2023)
- [A statistical framework for GWAS of high dimensional phenotypes using summary statistics, with application to metabolite GWAS](https://mlsanity.com/paper/2303.10221/) (2023)
- [Empirical Bayes Covariance Decomposition, and a Solution to the Multiple Tuning Problem in Sparse PCA](https://mlsanity.com/paper/2312.03274/) (2023)
- [Empirical Bayes Linked Matrix Decomposition](https://mlsanity.com/paper/2408.00237/) (2024)
- [Nash: Neural Adaptive Shrinkage for Structured High-Dimensional Regression](https://mlsanity.com/paper/2505.11143/) (2025)
- [On the Identifiability of Tensor Ranks via Prior Predictive Matching](https://mlsanity.com/paper/2510.14523/) (2025)

## Abstract (arXiv)

Matrix factorization methods - including Factor analysis (FA), and Principal Components Analysis (PCA) - are widely used for inferring and summarizing structure in multivariate data. Many matrix factorization methods exist, corresponding to different assumptions on the elements of the underlying matrix factors. For example, many recent methods use a penalty or prior distribution to achieve sparse representations ("Sparse FA/PCA"). Here we introduce a general Empirical Bayes approach to matrix factorization (EBMF), whose key feature is that it uses the observed data to estimate prior distributions on matrix elements. We derive a correspondingly-general variational fitting algorithm, which reduces fitting EBMF to solving a simpler problem - the so-called "normal means" problem. We implement this general algorithm, but focus particular attention on the use of sparsity-inducing priors that are uni-modal at 0. This yields a sparse EBMF approach - essentially a version of sparse FA/PCA - that automatically adapts the amount of sparsity to the data. We demonstrate the benefits of our approach through both numerical comparisons with competing methods and through analysis of data from the GTEx (Genotype Tissue Expression) project on genetic associations across 44 human tissues. In numerical comparisons EBMF often provides more accurate inferences than other methods. In the GTEx data, EBMF identifies interpretable structure that concords with known relationships among human tissues. Software implementing our approach is available at https://github.com/stephenslab/flashr