Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects

TL;DR

This paper develops a new inference method for non-negative matrix factorization with random effects, enabling covariate effect testing with uncertainty quantification via wild bootstrap, applicable to various data types.

Contribution

It introduces NMF-RE, a model combining covariate effects and heterogeneity, with a novel inference approach using wild bootstrap for covariate effect uncertainty quantification.

Findings

Wild bootstrap provides accurate uncertainty estimates for covariate effects.

Df control prevents model collapse and inference degeneracy.

Sparse loadings enable variable selection and interpretability.

Abstract

Non-negative matrix factorization (NMF) is widely used for parts-based representations, yet formal inference for covariate effects is rarely available when the basis is learned under non-negativity. We introduce non-negative matrix factorization with random effects (NMF-RE), a mean-structure latent-variable model $Y=X(\Theta A+U)+\mathcal{E}$ that combines covariate-driven scores with unit-specific deviations. Random effects act as a working device for modeling heterogeneity and controlling complexity; we monitor their effective degrees of freedom and enforce a df-based cap to prevent near-saturated fits. Estimation alternates closed-form ridge (BLUP-like) updates for $U$ with multiplicative non-negative updates for $X$ and $\Theta$. For inference on $\Theta$, we condition on $(\widehat X,\widehat U)$ and obtain fast uncertainty quantification via asymptotic linearization, a one-step Newton update, and a multiplier (wild) bootstrap; this avoids repeated constrained re-optimization. Simulations include a targeted stress test showing that, without df control, the random-effects penalty can collapse and inference for $\Theta$ becomes degenerate, whereas the df-cap prevents this failure mode. The non-negativity constraint induces sparse, parts-based loadings -- a measurement-side variable selection -- while inference on $\Theta$ identifies which covariates affect which components, providing covariate-side selection. Longitudinal, psychometric, spatial-flow, and text examples further illustrate stable, interpretable covariate-effect inference.