## 4 Applications to Center-Based Clustering
### Robustness of the Left-Sided Bregman Centroids  
Consider two \(d\)-dimensional points \(p=(p_{1},\ldots,p_{d})\) and \(p^{\prime}=(p^{\prime}_{1},\ldots,p^{\prime}_{d})\) of a domain \(\Theta\subset\mathbb{R}^{d}\). The centroid of \(p\) and \(p^{\prime}\) with respect to any arbitrary divergence \(D(\cdot:\cdot)\) is by definition the minimizer of  
\[L_{D}(c)=\frac{1}{2}D(p:c)+\frac{1}{2}D(p^{\prime}:c),\]  
provided that the minimizer \(\min_{c\in\Theta}L_{D}(c)\) is unique. Assume a separable Bregman divergence induced by the generator \(F(p)=\sum_{d=1}^{d}F(p_{i})\). The left-sided Bregman centroid [73] of \(p\) and \(p^{\prime}\) is given by the following separable quasi-arithmetic centroid:  
\[c=(c_{1},\ldots,c_{d}),\]with  
\[c_{i}=M^{f}\left(p_{i},p^{\prime}_{i}\right)=f^{-1}\left(\frac{f(p_{i})+f(p^{ \prime}_{i})}{2}\right),\]  
where \(f(x)=F^{\prime}(x)\) denotes the derivative of the Bregman generator \(F(x)\). Now, fix \(p\) (say, \(p=(1,\ldots,1)\in\Theta\)), and let the coordinates \(p^{\prime}_{i}\) of \(p^{\prime}\) all tend to infinity: That is, point \(p^{\prime}\) plays the role of an outlier data point. We use the general framework of influence functions [74] in statistics to study the robustness of divergence-based centroids. Consider the \(r\)-power mean, a quasi-arithmetic mean induced by \(\text{pow}_{r}(x)=x^{r}\) for \(r\
eq 0\) and by extension \(\text{pow}_{0}(x)=\log x\) when \(r=0\) (geometric mean). When \(r<0\), we check that  
\[\lim_{p^{\prime}_{i}\rightarrow+\infty}M^{\text{pow}_{r}}(p_{i}, p^{\prime}_{i}) = \lim_{p^{\prime}_{i}\rightarrow+\infty}\left(\frac{1+p^{\prime }_{i}}{2}\right)^{\frac{1}{r}}, \tag{109}\] \[= \left(\frac{1}{2}\right)^{\frac{1}{r}}<\infty. \tag{110}\]  
That is, the \(r\)-power mean is robust to an outlier data point when \(r<0\) (see Figure 2). Note that if instead of considering the centroid, we consider the barycenter with \(w\) denoting the weight of point \(p\) and \(1-w\) denoting the weight of the outlier \(p^{\prime}\) for \(w\in(0,1)\), then the power \(r\)-mean falls in a square box of side \(w^{\frac{1}{r}}\) when \(r<0\). On the contrary, when \(r>0\) or \(r=0\), we have \(\lim_{p^{\prime}_{i}\rightarrow+\infty}M^{\text{pow}_{r}}(p_{i},p^{\prime}_{i })=\infty\), and the \(r\)-power mean diverges to infinity. Thus, when \(r<0\), the quasi-arithmetic centroid of \(p=(1,\ldots,1)\) and \(p^{\prime}\) is contained in a bounding box of length \(\left(\frac{1}{2}\right)^{\frac{1}{r}}\) with left corner \((1,\ldots,1)\), and the left-sided Bregman power centroid minimizing  
\[\frac{1}{2}B_{F}(c:p)+\frac{1}{2}B_{F}(c:p^{\prime})\]  
is robust to outlier \(p^{\prime}\). To contrast with this result, notice that the right-sided Bregman centroid [72] is always the center of mass (arithmetic mean), and therefore not robust to outliers as a single outlier data point may potentially drag the centroid to infinity. Figure 2: Illustration of the robustness property of the \(r\)-power mean \(M^{\text{pow}_{r}}(p,p^{\prime})\) when \(r<0\) for two points: a prescribed point \(p=(1,1)\) and an outlier point \(p^{\prime}=(t,t)\). When \(t\rightarrow+\infty\), the \(r\)-power mean of \(p\) and \(p^{\prime}\) for \(r<0\) (e.g., coordinatewise harmonic mean when \(r=-1\)) is contained inside the box anchored at \(p\) of size length \(\left(\frac{1}{2}\right)^{\frac{1}{r}}\). The \(r\)-power mean can be interpreted as a left-sided Bregman centroid for \(F^{\prime}(x)=-x^{r}\), i.e., \(F(x)=-\frac{1}{r}x^{r+1}\) when \(r<-1\) and \(F(x)=-\log x\) when \(r=-1\). **Example 2**. : _Since \(M^{f}=M^{-f}\) for any strictly smooth increasing function \(f\), we deduce that the quasi-arithmetic left-sided Bregman centroid induced by \(F(x)=-\log x\) with \(f(x)=F^{\prime}(x)=-x^{-1}=-\frac{1}{x}\) for \(x>0\) is the harmonic mean which is robust to outliers.