Table 1.

TBD δ_f corresponding to f. $\stackrel{‒}{x}=1-x,\stackrel{‒}{y}=1-y,x^{\prime }$ is the transpose of x. Δ^d is d-simplex

X	f(x)	δf (x, y)	Remark
$\mathbb{R}$	x 2	(x − y)2 (1 + 4y2)−1/2	Total square loss
[0, 1]	$x\log x+\stackrel{‒}{x}\log \stackrel{‒}{x}$	$\left(x\log \frac{x}{y}+\stackrel{‒}{x}\log \frac{\stackrel{‒}{x}}{\stackrel{‒}{y}}\right)/\sqrt{1+y{(1+\log y)}^2+\stackrel{‒}{y}{(1+\log \stackrel{‒}{y})}^2}$	Total logistic loss
${\mathbb{R}}_{+}$	−log x	$\left(\frac{x}{y}-\log \frac{x}{y}-1\right)∕\sqrt{1+y^{-2}}$	Total Itakura-Saito distance
$\mathbb{R}$	ex	$(e^x-e^y-(x-y\left)e^y\right)∕\sqrt{1+e^{2y}}$
${\mathbb{R}}^d$	${\Vert x\Vert }^2$	${\Vert x-y\Vert }^2∕\sqrt{1+4{\Vert y\Vert }^2}$	Total squared Euclidean
${\mathbb{R}}^d$	x’Ax	$\left({(x-y)}^{\prime }A\right(x-y\left)\right)∕\sqrt{1+4{\Vert \mathit{Ay}\Vert }^2}$	Total Mahalanobis distance
Δ d	${\sum }_{j=1}^d{x}_j\log x_j$	$\left({\sum }_{j=1}^d{x}_j\log \frac{x_j}{y_j}\right)/\sqrt{1+{\sum }_{j=1}^d{y}_j{(1+\log y_j)}^2}$	Total KL divergence
${\mathbb{C}}^{m\times n}$	${\Vert x\Vert }_F^2$	${\Vert x-y\Vert }_F^2∕\sqrt{1+4{\Vert y\Vert }_F^2}$	Total squared Frobenius