Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects (original) (raw)
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(ThetaA+U)+mathcalEY=X(\Theta A+U)+\mathcal{E}Y=X(ThetaA+U)+mathcalE 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 UUU with multiplicative non-negative updates for XXX and Theta\ThetaTheta. For inference on Theta\ThetaTheta, we condition on (widehatX,widehatU)(\widehat X,\widehat U)(widehatX,widehatU) 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\ThetaTheta 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\ThetaTheta identifies which covariates affect which components, providing covariate-side selection. Longitudinal, psychometric, spatial-flow, and text examples further illustrate stable, interpretable covariate-effect inference.
Submission history
From: Kenichi Satoh [view email]
[v1] Mon, 2 Mar 2026 05:29:46 UTC (83 KB)