Machine Learning

, Volume 107, Issue 4, pp 795–795 | Cite as

Correction to: Semi-supervised AUC optimization based on positive-unlabeled learning

Correction
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1 Correction to: Mach Learn  https://doi.org/10.1007/s10994-017-5678-9

On page 7, 4.2 Variance reduction, the third line of the equation needs to be corrected as follows:
$$\begin{aligned} \sigma _\mathrm {NN}^2(f)={{\mathrm{Var}}}_{\mathrm {N}{\bar{\mathrm{N}}}} [\ell (f(\varvec{x}^{\mathrm {N}},\varvec{\bar{x}}^{\mathrm {N}}))], \end{aligned}$$
Footnote 4 needs the following correction:

\({{\mathrm{Var}}}_{\mathrm {PN}}\), \({{\mathrm{Var}}}_{\mathrm {P}\bar{\mathrm{P}}}\), and \({{\mathrm{Var}}}_{\mathrm {N}\bar{\mathrm{N}}}\) are the variances over \(p_{{\mathrm {P}}}(\varvec{x}^{\mathrm {P}})p_{{\mathrm {N}}}(\varvec{x}^{\mathrm {N}})\), \(p_{{\mathrm {P}}}(\varvec{x}^{\mathrm {P}})p_{{\mathrm {P}}}(\varvec{\bar{x}}^{\mathrm {P}})\), and \(p_{{\mathrm {N}}}(\varvec{x}^{\mathrm {N}})p_{{\mathrm {N}}}(\varvec{\bar{x}}^{\mathrm {N}})\), respectively. \({{\mathrm{Cov}}}_{\mathrm {PN,P}\bar{\mathrm{P}}}\), \({{\mathrm{Cov}}}_{\mathrm {PN,N}\bar{\mathrm{N}}}\), \({{\mathrm{Cov}}}_{\mathrm {PU,P}\bar{\mathrm{P}}}\), and \({{\mathrm{Cov}}}_{\mathrm {NU,N}\bar{\mathrm{N}}}\) are the covariances over \(p_{{\mathrm {P}}}(\varvec{x}^{\mathrm {P}})p_{{\mathrm {N}}}(\varvec{x}^{\mathrm {N}})p_{{\mathrm {P}}}(\varvec{\bar{x}}^{\mathrm {P}})\), \(p_{{\mathrm {P}}}(\varvec{x}^{\mathrm {P}})p_{{\mathrm {N}}}(\varvec{x}^{\mathrm {N}})p_{{\mathrm {N}}}(\varvec{\bar{x}}^{\mathrm {N}})\), \(p_{{\mathrm {P}}}(\varvec{x}^{\mathrm {P}})p(\varvec{x}^{\mathrm {U}})p_{{\mathrm {P}}}(\varvec{\bar{x}}^{\mathrm {P}})\), and \(p_{{\mathrm {N}}}(\varvec{x}^{\mathrm {N}})p(\varvec{x}^{\mathrm {U}})p_{{\mathrm {N}}}(\varvec{\bar{x}}^{\mathrm {N}})\), respectively.

On page 26, Appendix C: Proof of variance reduction, the last equation needs correction:
$$\begin{aligned} {{\mathrm{Var}}}[\widehat{R}_\mathrm {PNNU}^{\gamma }(g)]&= \frac{(1-\gamma )^2}{{n_{{\mathrm {P}}}}{n_{{\mathrm {N}}}}}\sigma _\mathrm {PN}^2(g) +\frac{\gamma ^2\theta _{{\mathrm {N}}}^2}{{n_{{\mathrm {N}}}}^2\theta _{{\mathrm {P}}}^2}\sigma _\mathrm {NN}^2(g) +\frac{(1-\gamma )\gamma }{\theta _{{\mathrm {P}}}{n_{{\mathrm {N}}}}}\tau _\mathrm {PN,NU}(g) \\&\phantom {=}- \frac{\gamma ^2\theta _{{\mathrm {N}}}}{\theta _{{\mathrm {P}}}^2{n_{{\mathrm {N}}}}}\tau _\mathrm {NU,NN}(g) -\frac{(1-\gamma )\gamma \theta _{{\mathrm {N}}}}{\theta _{{\mathrm {P}}}{n_{{\mathrm {N}}}}} \tau _\mathrm {PN,NN}(g) \\&=(1-\gamma )^2 \psi _\mathrm {PN}+ \gamma ^2 \psi _\mathrm {NU}+ (1-\gamma )\gamma \psi _\mathrm {NN}. \end{aligned}$$
The original article has been corrected.

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Center for Advanced Intelligence ProjectRIKENTokyoJapan
  2. 2.Graduate School of Frontier SciencesThe University of TokyoChibaJapan

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