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Context: Moreover, any regular exponential family distribution p_{\psi}(\mathbf{z}|\bm{\theta}) with parameters \bm{\theta} and cumulant function \psi can be written in terms of a uniquely determined regular Bregman divergence Banerjee et al. (2005):p_{\psi}(\mathbf{z}|\bm{\theta})=\exp\{\mathbf{z}^{T}\bm{\theta}-\psi(\bm{\theta})-g_{\psi}(\mathbf{z})\}=\exp\{-d_{\varphi}(\mathbf{z},\bm{\mu}(\bm{\theta}))-g_{\varphi}(\mathbf{z})\}(4)Consider now a regular exponential family mixture model with parameters \bm{\Gamma}=\{\bm{\theta}_{k},\pi_{k}\}_{k=1}^{K}:p(\mathbf{z}|\bm{\Gamma})=\sum_{k=1}^{K}\pi_{k}p_{\psi}(\mathbf{z}|\bm{\theta}_{k})=\sum_{k=1}^{K}\pi_{k}\exp(-d_{\varphi}(\mathbf{z},\bm{\mu}(\bm{\theta}_{k}))-g_{\varphi}(\mathbf{z}))(5)Given \bm{\Gamma}, inference of the cluster assignment y for an unlabeled point \mathbf{z} becomes:p(y=k|\mathbf{z})=\frac{\pi_{k}\exp(-d_{\varphi}(\mathbf{z},\bm{\mu}(\bm{\theta}_{k})))}{\sum_{k^{\prime}}\pi_{k^{\prime}}\exp(-d_{\varphi}(\mathbf{z},\bm{\mu}(\bm{\theta}_{k})))}(6)For an equally-weighted mixture model with one cluster per class, cluster assignment inference (6) is equivalent to query class prediction (2) with f_{\phi}(\mathbf{x})=\mathbf{z} and \mathbf{c}_{k}=\bm{\mu}(\bm{\theta}_{k}).In this case, prototypical networks are effectively performing mixture density estimation with an exponential family distribution determined by d_{\varphi}. The choice of distance therefore specifies modeling assumptions about the class-conditional data distribution in the embedding space.

Question: Is it true that any regular exponential family distribution pψ(z|θ) with parameters θ and cumulant function ψ can be written in terms of a uniquely determined regular Bregman divergence?
[Yes, any regular exponential family distribution p_{\psi}(\mathbf{z}|\bm{\theta}) with parameters \bm{\theta} and cumulant function \psi can be written in terms of a uniquely determined regular Bregman divergence as: p_{\psi}(\mathbf{z}|\bm{\theta})=\exp\{\mathbf{z}^{T}\bm{\theta}-\psi(\bm{\theta})-g_{\psi}(\mathbf{z})\}=\exp\{-d_{\varphi}(\mathbf{z},\bm{\mu}(\bm{\theta}))-g_{\varphi}(\mathbf{z})\}]