Finite Poisson NMF multiplicative-update sweeps

Input an admissible count matrix $X=(x_{ij})\in {\mathbb{N}}_0^{n\times p}$ with no all-zero row or column, an inner dimension $r\ge 1$, a finite sweep count $T\in {\mathbb{N}}_0$, and strictly positive initial factors $W^{(0)}\in (0,\mathrm{\infty }{)}^{n\times r}$ and $H^{(0)}\in (0,\mathrm{\infty }{)}^{r\times p}$. For each $t=0,\dots ,T-1$, perform one complete sweep in the stated order:

1. Set ${\lambda }_{ij}^{(t)}={\sum }_{k=1}^r{W}_{ik}^{(t)}{H}_{kj}^{(t)}$.

2. For every $i,k$, set

   $W_{ik}^{(t+1)}=W_{ik}^{(t)}\frac{{\sum }_{j=1}^p{H}_{kj}^{(t)}{x}_{ij}\mathrm{/}{\lambda }_{ij}^{(t)}}{{\sum }_{j=1}^p{H}_{kj}^{(t)}}\mathrm{.}$

3. Recompute ${\tilde{\lambda }}_{ij}^{(t)}={\sum }_{k=1}^r{W}_{ik}^{(t+1)}{H}_{kj}^{(t)}$.

4. For every $k,j$, set

   $H_{kj}^{(t+1)}=H_{kj}^{(t)}\frac{{\sum }_{i=1}^n{W}_{ik}^{(t+1)}{x}_{ij}\mathrm{/}{\tilde{\lambda }}_{ij}^{(t)}}{{\sum }_{i=1}^n{W}_{ik}^{(t+1)}}\mathrm{.}$

The first factor block uses the current $H^{(t)}$; the second uses the new $W^{(t+1)}$ and the recomputed intermediate rates.

Return $W^{(T)},H^{(T)}$. The local well-definedness result proves that every denominator and rate in these finite sweeps is positive under the stated assumptions. This algorithm makes no likelihood-monotonicity, convergence, stationary-point, or global-optimality claim.

Poisson nonnegative matrix-factorization model

Independent Poisson counts have the stated NMF log-likelihood

Strictly positive initialization keeps the Poisson NMF updates well-defined ↗

Hastie, Tibshirani, and Friedman, §14.6

Lee and Seung, §4