###### Abstract  
The family of \(\alpha\)-divergences including the oriented forward and reverse Kullback-Leibler divergences is often used in signal processing, pattern recognition, and machine learning, among others. Choosing a suitable \(\alpha\)-divergence can either be done beforehand according to some prior knowledge of the application domains or directly learned from data sets. In this work, we generalize the \(\alpha\)-divergences using a pair of strictly comparable weighted means. Our generalization allows us to obtain in the limit case \(\alpha\to 1\) the 1-divergence, which provides a generalization of the forward Kullback-Leibler divergence, and in the limit case \(\alpha\to 0\), the 0-divergence, which corresponds to a generalization of the reverse Kullback-Leibler divergence. We then analyze the condition for a pair of weighted quasi-arithmetic means to be strictly comparable and describe the family of quasi-arithmetic \(\alpha\)-divergences including its subfamily of power homogeneous \(\alpha\)-divergences. In particular, we study the generalized quasi-arithmetic 1-divergences and 0-divergences and show that these counterpart generalizations of the oriented Kullback-Leibler divergences can be rewritten as equivalent conformal Bregman divergences using strictly monotone embeddings. Finally, we discuss the applications of these novel divergences to \(k\)-means clustering by studying the robustness property of the centroids. K +
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Since the KLD is independent of the reference measure \(\mu\), i.e., \(\mathrm{KL}_{\mu}(p_{\mu}:q_{\mu})=\mathrm{KL}_{\
u}(p_{\
u}:q_{\
u})\) for \(p_{\mu}=\frac{\mathrm{d}p}{\mathrm{d}\mu}\) and \(q_{\mu}=\frac{\mathrm{d}Q}{\mathrm{d}\mu}\), and \(p_{\
u}=\frac{\mathrm{d}P}{\mathrm{d}\
u}\) and \(q_{\
u}=\frac{\mathrm{d}Q}{\mathrm{d}\
u}\) are the RN derivatives with respect to another positive measure \(\
u\), we write concisely in the remainder:  
\[\mathrm{KL}(p:q)=\int p\log\frac{p}{q}\mathrm{d}\mu, \tag{2}\]  
instead of \(\mathrm{KL}_{\mu}(p_{\mu}:q_{\mu})\). The KLD belongs to a parametric family of \(\alpha\)-divergences [9]\(I_{\alpha}(p:q)\) for \(\alpha\in\mathbb{R}\):  
\[I_{\alpha}(p:q):=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}\big{(}1 -\int p^{\alpha}q^{1-\alpha}\mathrm{d}\mu\big{)},&\alpha\in\mathbb{R}\backslash \{0,1\}\\ I_{1}(p:q)=\mathrm{KL}(p:q),&\alpha=1\\ I_{0}(p:q)=\mathrm{KL}(q:p),&\alpha=0\end{array} \right. \tag{3}\]  
The \(\alpha\)-divergences extended to positive densities [10] (not necessarily normalized densities) play a central role in information geometry [6]:  
\[I_{\alpha}^{+}(p:q):=\left\{ \begin{array}{ll}\frac{1}{\alpha(1-\alpha)}\int \big{(}ap+(1-\alpha)q-p^{\alpha}q^{1-\alpha}\big{)}\mathrm{d}\mu,&\alpha\in \mathbb{R}\backslash\{0,1\}\\ I_{1}^{+}(p:q)=\mathrm{KL}^{+}(p:q),&\alpha=1\\ I_{0}^{+}(p:q)=\mathrm{KL}^{+}(q:p),&\alpha=0\end{array} \right., \tag{4}\]  
where \(\mathrm{KL}^{+}\) denotes the Kullback-Leibler divergence extended to positive measures:  
\[\mathrm{KL}^{+}(p:q):=\int\bigg{(}p\log\frac{p}{q}+q-p\bigg{)}\mathrm{d}\mu. \tag{5}\]  
The \(\alpha\)-divergences are asymmetric for \(\alpha\
eq\frac{1}{2}\) (i.e., \(I_{\alpha}(p:q)\
eq I_{\alpha}(q:p)\) for \(\alpha\
eq\frac{1}{2}\)) but exhibit the following reference duality [11]:  
\[I_{\alpha}(q:p)=I_{1-\alpha}(p:q)=:I_{\alpha}^{*}(p:q), \tag{6}\]  
where we denoted by \(D^{*}(p:q):=D(q:p)\), the reverse divergence for an arbitrary divergence \(D(p:q)\) (e.g., \(I_{\alpha}^{*}(p:q):=I_{\alpha}(q:p)=I_{1-\alpha}(p:q)\)). The \(\alpha\)-divergences have been extensively used in many applications [12], and the parameter \(\alpha\) may not be necessarily fixed beforehand but can also be learned from data sets in applications [13; 14].