NMF-FFB: Non-negative matrix factorization with feedforward-feedback structure

Non-negative matrix factorization (NMF) approximates a non-negative endogenous data matrix as $Y_1 \approx XB$, with non-negative latent components $X$ and coefficients $B$. Standard covariate-aware NMF is feedforward: $B$ depends only on exogenous variables $Y_2$, with no latent feedback among endogenous variables. We propose NMF-FFB (NMF with feedforward-feedback structure), an exploratory data-fitting framework that embeds the simultaneous equation $B = \Theta_1 Y_1 + \Theta_2 Y_2$ in NMF, where $\Theta_1$ is non-negative latent feedback and $\Theta_2$ non-negative exogenous pathways. NMF-FFB is positioned within data-fitting structural equation modeling (SEM): it fits $Y_1$ directly rather than a model-implied covariance, and is not a confirmatory measurement model or a replacement for maximum-likelihood SEM under standard confirmatory factor analysis assumptions. When $\rho(X\Theta_1)<1$, the reduced form $Y_1 \approx (I-X\Theta_1)^{-1} X\Theta_2 Y_2$ defines a latent Leontief inverse separating direct from cumulative feedback-amplified effects. Estimation uses regularized multiplicative updates with orthogonality and sparsity penalties; an $X$-fixed bootstrap summarizes uncertainty for the feedback spectral radius, the amplification ratio, and path coefficients. Unlike conventional SEM, NMF-FFB requires only the latent rank $Q$ and lets $X$ group endogenous indicators into latent factors. This suits non-negative additive data, automatic loading discovery, Leontief-type cumulative effects, and small samples where covariance-based maximum-likelihood fitting is ill-conditioned. Applications to Holzinger-Swineford, Los Angeles pollution-mortality, and Mississippi county-level health data demonstrate interpretable parts-based representations across distinct latent-feedback regimes.