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Covariate-moderated Empirical Bayes Matrix Factorization
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Proposes covariate-moderated empirical Bayes matrix factorization (cEBMF), which makes EBMF priors depend on row/column side information via any probabilistic model or neural network, fit by a coordinate-ascent variational EB algorithm.
Problem
Matrix factorization with side information (images, text, graphs, covariates) is limited by restrictive parametric models and fixed priors that often don't fit the data, and existing methods rarely handle arbitrary covariate types or adapt priors to data.
Approach
Extends EBMF by making each factor's prior a parameterized distribution g(x_i) whose parameters (e.g., mixture weights via logistic/softmax regression or an MLP) depend on row/column covariates, fit via a covariate-moderated EBNM subproblem inside a variational coordinate-ascent algorithm, with hyperparameters chosen by empirical Bayes.
Key results
- In simulations, cEBMF was generally more accurate than EBMF and other methods, with largest gains in the informative tiled-clustering setting and no worse than EBMF under uninformative or misspecified covariates.
- On MovieLens 100K, cEBNMF yielded overall the best prediction accuracy across training-set splits, outperforming EBNMF and deep learning baselines.
- cEBMF scaled to n = 1,000,000 rows (3,403.6 s), while MFAI failed beyond 100,000 rows and Spatial PCA needed about 300 GB at n = 100,000.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Incorporate covariates through learned covariate-dependent priors | cEBMF versus EBMF · Simulations with informative covariates, especially tiled-clustering | RMSE | Helped |
| Incorporate uninformative covariates through learned priors | cEBMF versus EBMF · Uninformative covariate simulations | RMSE | No clear effect |
| Use covariate-dependent priors under prior misspecification | cEBMF versus EBMF · Shifted tiled-clustering simulations | RMSE | No clear effect |
| Use covariates in the conditional VAE | cVAE versus VAE · Sparsity-driven simulations | RMSE | No clear effect |
| Use covariates in the conditional VAE | cVAE versus VAE · Tiled-clustering simulations | RMSE | Helped |
| Use Spatial PCA's spatial side-information model | Spatial PCA versus EBMF · Tiled-clustering and shifted tiled-clustering simulations | RMSE | Hurt |
| Use MFAI instead of EBMF | MFAI · Simulations, except shifted tiled-clustering | RMSE | Hurt |
| Incorporate movie genres into non-negative factorization priors | cEBNMF versus EBNMF · MovieLens 100K | RMSE | Helped |
| Incorporate movie genres into factorization priors | MFAI and cEBNMF versus EBNMF · MovieLens 100K | RMSE | Helped |
| Use covariate-dependent priors for the non-negative factors | cEBNMF versus EBNMF · DLPFC spatial transcriptomics | spatial alignment with expert labeling | Helped |
| Condition the VAE on spatial side information | cVAE versus VAE · DLPFC spatial transcriptomics | spatial structure and expert-label alignment | No clear effect |
| Use covariate-moderated priors rather than unmoderated priors | cEBMF versus EBMF · Scalability benchmark, n=10^3 to n=10^6 | running time (seconds): 0.8 -> 5.2; 2.5 -> 35.5; 36.9 -> 416.3; 165.1 -> 3,403.6 | Hurt |
Surprising findings
- cVAE did not improve over VAE on spatial transcriptomics despite using side information.
- Spatial PCA underperformed EBMF, which does not use side information, in tiled-clustering simulations.
Limitations
- Relies on priors that may be misspecified; performance depends on prior family choice (shown in shifted tiled-clustering experiment).
- Computationally slower than plain EBMF (about 10x in scalability benchmark).
- Experiments use only small-scale real datasets (MovieLens 100K, one DLPFC dataset set) and the paper does not evaluate image or graph covariates.
Experimental setup
Datasets: MovieLens 100K DLPFC spatial transcriptomics (12 slices) simulated matrices
Benchmarks: MovieLens 100K DLPFC spatial transcriptomics
Metrics: RMSE running time
Baselines: EBMF PMD VAE MFAI Spatial PCA cVAE NCF NMF (NNLM) EBNMF
Builds on: empirical Bayes matrix factorization (EBMF) flashier empirical Bayes normal means (EBNM) MFAI SpatialPCA
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