TABLE 1.

Instances of Bregman Divergence and Point-to-Centroid Distance

Distance	Domain	$\varphi (\mathbf{x})$	$f(\mathbf{x},\mathbf{y})$
Logistic loss	{0, 1}	$x\log x$	$x\log (\frac{x}{y})-(x-y)$
Itakura-Saito distance	${\mathbf{R}}_{++}$	$-\log x$	$\frac{x}{y}-\log (\frac{x}{y})-1$
Squared Euclidean distance	${\mathbf{R}}^d$	${\Vert \mathbf{x}\Vert }_2^2$	${\Vert \mathbf{x}-\mathbf{y}\Vert }_2^2$
Mahalanobis distance	${\mathbf{R}}^d$	$\mathbf{x}\mathbf{A}\mathbf{x}$	$(\mathbf{x}-\mathbf{y})\mathbf{A}{(\mathbf{x}-\mathbf{y})}^{⊺}$
KL-divergence	$d$-Simplex	${\sum }_{j=1}^d{x}_j\log x_j$	${\sum }_{j=1}^d{x}_j\log (\frac{x_j}{y_j})$
Generalized I-divergence	${\mathbf{R}}_{+}^d$	${\sum }_{j=1}^d{x}_j\log x_j$	${\sum }_{j=1}^d{x}_j\log (\frac{x_j}{y_j})-{\sum }_{j=1}^d(x_j-y_j)$
Cosine similarity	${\mathbf{R}}^d$	${\Vert \mathbf{x}\Vert }_2$	${\Vert \mathbf{x}\Vert }_2-{\sum }_{j=1}^d{x}_j{y}_j∕{\Vert \mathbf{y}\Vert }_2$

Note: $\mathbf{A}$ is a $d\times d$ inverse of the covariance matrix.