NMF-FFB: Non-negative matrix factorization with feedforward-feedback structure
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arXiv
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| Format: | Preprint |
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2025
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| author | Satoh, Kenichi |
| author_facet | Satoh, Kenichi |
| contents | 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 $B = Θ_1 Y_1 + Θ_2 Y_2$ in NMF, where $Θ_1$ is non-negative latent feedback and $Θ_2$ 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 $ρ(XΘ_1)<1$, the reduced form $Y_1 \approx (I-XΘ_1)^{-1} XΘ_2 Y_2$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | NMF-FFB: Non-negative matrix factorization with feedforward-feedback structure Satoh, Kenichi Methodology 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 $B = Θ_1 Y_1 + Θ_2 Y_2$ in NMF, where $Θ_1$ is non-negative latent feedback and $Θ_2$ 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 $ρ(XΘ_1)<1$, the reduced form $Y_1 \approx (I-XΘ_1)^{-1} XΘ_2 Y_2$ 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. |
| title | NMF-FFB: Non-negative matrix factorization with feedforward-feedback structure |
| topic | Methodology |
| url | https://arxiv.org/abs/2512.18250 |