Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects
arXiv:2603.01468v1 Announce Type: cross
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.
