Strictly positive initialization keeps the Poisson NMF updates well-defined

Since $W,H$ are strictly positive, every current rate ${\lambda }_{ij}={\sum }_k{W}_{ik}{H}_{kj}$ is positive. For each row $i$, choose $j_0$ with $x_{i{j}_0}>0$ using [F1]. Then for every $k$, the numerator ${\sum }_j{H}_{kj}{x}_{ij}\mathrm{/}{\lambda }_{ij}$ has the positive term $H_{k{j}_0}{x}_{i{j}_0}\mathrm{/}{\lambda }_{i{j}_0}$, while its denominator ${\sum }_j{H}_{kj}$ is positive. Thus every updated $W_{ik}$ is positive.

The intermediate rates ${\tilde{\lambda }}_{ij}={\sum }_k{W}_{ik}^{\mathrm{n}\mathrm{e}\mathrm{w}}{H}_{kj}$ are positive. For each column $j$, choose $i_0$ with $x_{i_{0}j}>0$ using [F1]. For every $k$, the numerator ${\sum }_i{W}_{ik}^{\mathrm{n}\mathrm{e}\mathrm{w}}{x}_{ij}\mathrm{/}{\tilde{\lambda }}_{ij}$ has the positive term at $i_0$, and its denominator ${\sum }_i{W}_{ik}^{\mathrm{n}\mathrm{e}\mathrm{w}}$ is positive. Thus every updated $H_{kj}$ is positive. Repeating this argument inductively proves the claim for each finite sweep.