Table 1:

Different losses and their loss-specific centers. We provide all calculations associated with loss-specific centers in Appendix F. Note the Gecco problem with Hamming or Canberra distances is not convex. Though we discuss general convex losses in this paper, we list these non-convex losses for reference. For multinomial log-likelihood and multinomial deviance, we change Gecco formulation slightly to accommodate three indices; we provide a detailed formulation in Appendix E.

Data Type	Loss Type	Loss Function	Loss-specific Center $\tilde{\mathbf{x}}$
Continuous	Euclidean (ℓ2)	$\frac{1}{2}{\Vert {\mathbf{\text{x}}}_i-{\mathbf{\text{u}}}_i\Vert }_2^2$	$\overline{\mathbf{x}}$
Skewed continuous	Manhattan (ℓ1) Minkowski (ℓq) Mahalanobis (weighted ℓ2) Chebychev (ℓ∞) Canberra (weighted ℓ1)	${\sum }_{j=1}\left|x_{ij}-u_{ij}\right|$ $\sqrt[q]{{\sum }_{j=1}{\left|x_{ij}-u_{ij}\right|}^q}$ ${({\mathbf{\text{x}}}_i-{\mathbf{\text{u}}}_i)}^T{\mathbf{\text{C}}}^{-1}({\mathbf{\text{x}}}_i-u_i)$ ${\max }_j\left\{\left|x_{ij}-u_{ij}\right|\right\}$ ${\sum }_{j=1}\frac{\left|x_{ij}-u_{ij}\right|}{\left|x_{ij}\right|+\left|u_{ij}\right|}$	median(x) no closed form no closed form no closed form no closed form
Binary	Bernoulli log-likelihood Binomial deviance Hinge loss KL divergence Hamming (ℓ0)	$-x_{ij}{u}_{ij}+\log (1+e^{u_{ij}})$ $-x_{ij}\log u_{ij}-(1-x_{ij})\log (1-u_{ij})$ $\max (0,1-u_{ij}{x}_{ij})$ $-x_{ij}{\log }_2{u}_{ij}$ ${\sum }_j\#(x_{ij}\ne u_{ij})/n$	logit($\overline{\mathbf{x}}$) $\overline{\mathbf{x}}$ mode(x) no closed form mode (x)
Count	Poisson log-likelihood Poisson deviance Negative binomial log-likelihood Negative binomial deviance Manhattan (ℓ1) Canberra (weighted ℓ1)	$-x_{ij}{u}_{ij}+\exp (u_{ij})$ $-x_{ij}\log u_{ij}+u_{ij}$ $-x_{ij}{u}_{ij}+(x_{ij}+\frac{1}{\alpha })\log (\frac{1}{\alpha }+e^{u_{ij}})$ $x_{ij}\log (\frac{x_{ij}}{u_{ij}})-(x_{ij}+\frac{1}{\alpha })\log (\frac{1+\alpha x_{ij}}{1+\alpha u_{ij}})$ ${\sum }_{j=1}\left|x_{ij}-u_{ij}\right|$ ${\sum }_{j=1}\frac{\left|x_{ij}-u_{ij}\right|}{\left|x_{ij}\right|+\left|u_i\right|}$	log($\overline{\mathbf{x}}$) $\overline{\mathbf{x}}$ log($\overline{\mathbf{x}}$) $\overline{\mathbf{x}}$ median(x) no closed form
Categorical	Multinomial log-likelihood Multinomial deviance	$\left\{{\sum }_{k=1}^K-x_{ijk}{u}_{ijk}+\log ({\sum }_{k=1}^K{e}^{u_{ijk}})\right\}$ $\left\{{\sum }_{k=1}^K-x_{ijk}+\log (u_{ijk})\right\},\sum _{k=1}^K{u}_{ijk}=1$	mlogit($\overline{\mathbf{x}}$) $\overline{\mathbf{x}}$