NMF-FFB: Non-negative matrix factorization with feedforward-feedback structure (original) (raw)
Abstract:Non-negative matrix factorization (NMF) approximates a non-negative endogenous data matrix as Y1approxXBY_1 \approx XBY1approxXB, with non-negative latent components XXX and coefficients BBB. Standard covariate-aware NMF is feedforward: BBB depends only on exogenous variables Y2Y_2Y2, 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=Theta1Y1+Theta2Y2B = \Theta_1 Y_1 + \Theta_2 Y_2B=Theta1Y1+Theta2Y2 in NMF, where Theta1\Theta_1Theta1 is non-negative latent feedback and Theta2\Theta_2Theta2 non-negative exogenous pathways. NMF-FFB is positioned within data-fitting structural equation modeling (SEM): it fits Y1Y_1Y1 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(XTheta1)<1\rho(X\Theta_1)<1rho(XTheta1)<1, the reduced form Y1approx(I−XTheta1)−1XTheta2Y2Y_1 \approx (I-X\Theta_1)^{-1} X\Theta_2 Y_2Y_1approx(I−XTheta_1)−1XTheta_2Y_2 defines a latent Leontief inverse separating direct from cumulative feedback-amplified effects. Estimation uses regularized multiplicative updates with orthogonality and sparsity penalties; an XXX-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 QQQ and lets XXX 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.
Submission history
From: Kenichi Satoh [view email]
[v1] Sat, 20 Dec 2025 07:22:44 UTC (242 KB)
[v2] Fri, 15 May 2026 09:03:18 UTC (124 KB)