Poisson EPCA and the Generalized KL-Divergence

The cumulant of the Poisson distribution is $G(\theta) = e^\theta$, so the the link function (its derivative) is $g(\theta) = e^\theta$. From the appendix, we know that $f(x) = g^{-1}(x) = -\log{x}$ and

\[\begin{aligned} F(x) &= \theta \cdot x - G(\theta) \\ &= f(x) \cdot x - G(f(x)) \\ &= x \log{x} - x. \end{aligned}\]

The Bregman divergence induced from $F$ is

\[\begin{aligned} B_F(p \| q) &= F(p) - F(q) - \langle f(q), p - q \rangle \\ &= p \log p - p - q \log q + q - \langle \log q, p - q \rangle \\ &= p \log \frac{p}{q} - p + q. \end{aligned}\]

so $B_F$ is generalized KL-divergence.