## 2 The \(\alpha\)-Divergences Induced by a Pair of Strictly Comparable Weighted Means
### Generalized KL Divergences as Conformal Bregman Divergences on Monotone Embeddings  
Let us rewrite the generalized KLDs \(I_{1}^{f,\mathcal{S}}\) as a conformal Bregman representational divergence [58; 59; 60] as follows:  
**Theorem 4**. : _The generalized KLDs \(I_{1}^{f,\mathcal{S}}\) divergences are conformal Bregman representational divergences_  
\[I_{1}^{f,\mathcal{S}}(p:q)=\int\frac{1}{f^{\prime}(p)}\ B_{F}(g(q):g(p)) \mathrm{d}\mu, \tag{77}\]  
_with \(F=f\circ g^{-1}\) a strictly convex and differentiable Bregman convex generator defining the scalar Bregman divergence [61]\(B_{F}\):_  
\[B_{F}(a:b)=F(a)-F(b)-(a-b)F^{\prime}(b).\]  
Proof. : For the Bregman strictly convex and differentiable generator \(F=f\circ g^{-1}\), we expand the following conformal divergence  
\[\frac{1}{f^{\prime}(p)}B_{F}(g(q):g(p)) = \frac{1}{f^{\prime}(p)}\big{(}F(g(q))-F(g(p))-(g(q)-g(p))F^{ \prime}(g(p))  
In general, for a functional generator \(f\) and a strictly monotonic representational function \(r\) (also called monotone embedding [62] in information geometry), we can define the representational Bregman divergence [63]\(B_{for^{-1}}(r(p):r(q))\) provided that \(F=f\circ r^{-1}\) is a Bregman generator (i.e., strictly convex and differentiable).  
The Itakura-Saito divergence [64] (IS) between two densities \(p\) and \(q\) is defined by:  
\[D_{\text{IS}}(p:q) = \int\biggl{(}\frac{p}{q}-\log\frac{p}{q}-1\biggr{)}\text{d}\mu, \tag{82}\] \[= \int D_{\text{IS}}(p(x):q(x))\text{d}\mu(x), \tag{83}\]  
where \(D_{\text{IS}}(x:y)=\frac{x}{y}-\log\frac{x}{y}-1\) is the scalar IS divergence. This divergence was originally designed in sound processing for measuring the discrepancy between two speech power spectra. Observe that the IS divergence is invariant by rescaling: \(D_{\text{IS}}(tp:tq)=D_{\text{IS}}(p:q)\) for any \(t>0\). The IS divergence is a Bregman divergence [61] obtained for the Burg information generator (i.e., negative Burg entropy): \(F_{\text{Burg}}(u)=-\log u\) with \(F^{f}_{\text{Burg}}(u)=-\frac{1}{u}\). It follows that we have  
\[I^{f}_{1}(p:q)=\int pB_{f}(q:p)\text{d}\mu, \tag{84}\]  
The Itakura-Saito divergence may further be extended to a family of \(\alpha\)-Itakura-Saito divergences (see [6], Equation (10).45 of Theorem 10.1):  
\[D_{\text{IS},\alpha}(p:q) = \left\{ \begin{array}{ll}\int\frac{1}{a^{2}}\left(\left(\frac{p }{q}\right)^{\alpha}-\alpha\log\frac{p}{q}-1\right)\text{d}\mu&\alpha\
eq 0\\ \frac{1}{2}\int(\log q-\log p)^{2}\text{d}\mu&\alpha=0.\end{array} \right. \tag{85}\]  
In [56], a generalization of the Bregman divergences was obtained using the comparative convexity induced by two abstract means \(M\) and \(N\) to define \((M,N)\)-Bregman divergences as limit of scaled \((M,N)\)-[PERSON] divergences. The skew \((M,N)\)-Jensen divergences are defined for \(\alpha\in(0,1)\) by:  
\[J^{M,N}_{F,\alpha}(p:q)=\frac{1}{\alpha(1-\alpha)}(N_{\alpha}(F(p),F(q)))-F(M _{\alpha}(p,q))), \tag{86}\]  
where \(M_{\alpha}\) and \(N_{\alpha}\) are weighted means that should be regular [56] (i.e., homogeneous, symmetric, continuous, and increasing in each variable). Then, we can define the \((M,N)\)-Bregman divergence as  
\[B^{M,N}_{F}(p:q) = \lim_{\alpha\to 1^{-}}J^{M,N}_{F,\alpha}(p:q), \tag{87}\] \[= \lim_{\alpha\to 1^{-}}\frac{1}{\alpha(1-\alpha)}(N_{\alpha}(F(p),F (q)))-F(M_{\alpha}(p,q))). \tag{88}\]  
The formula obtained in [56] for the quasi-arithmetic means \(M^{f}\) and \(M^{g}\) and a functional generator \(F\) that is \((M^{f},M^{g})\)-convex is:  
\[B^{f,g}_{F}(p:q) = \frac{g(F(p))-g(F(q))}{g^{\prime}(F(q))}-\frac{f(p)-f(q)}{f^{ \prime}(q)}F^{\prime}(q), \tag{89}\] \[= \frac{1}{f^{\prime}(F(q))}B_{g\circ F\circ f^{-1}}(f(p):f(q))\geq 0. \tag{90}\]  
This is a conformal divergence [58] that can be written using the \(E_{f}\) terms as:  
\[B^{f,g}_{F}(p:q)=E_{g}(F(q),F(p))-E_{f}(q,p)F^{\prime}(q). \tag{91}\]  
A function \(F\) is \((M^{f},M^{g})\)-convex iff \(g\circ F\circ f^{-1}\) is (ordinary) convex [56].