Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects
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 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 with multiplicative non-negative updates for and . For inference on , we condition on 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 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 identifies which covariates affect which components, providing covariate-side selection. Longitudinal, psychometric, spatial-flow, and text examples further illustrate stable, interpretable covariate-effect inference.
Cite
@article{arxiv.2603.01468,
title = {Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects},
author = {Kenichi Satoh},
journal= {arXiv preprint arXiv:2603.01468},
year = {2026}
}