In domain adaptation, covariate shift and label shift problems are two
distinct and complementary tasks. In covariate shift adaptation where the
differences in data distribution arise from variations in feature
probabilities, existing approaches naturally address this problem based on
\textit{feature probability matching} (\textit{FPM}). However, for label shift
adaptation where the differences in data distribution stem solely from
variations in class probability, current methods still use FPM on the
$d$-dimensional feature space to estimate the class probability ratio on the
one-dimensional label space. To address label shift adaptation more naturally
and effectively, inspired by a new representation of the source domain's class
probability, we propose a new framework called \textit{class probability
matching} (\textit{CPM}) which matches two class probability functions on the
one-dimensional label space to estimate the class probability ratio,
fundamentally different from FPM operating on the $d$-dimensional feature
space. Furthermore, by incorporating the kernel logistic regression into the
CPM framework to estimate the conditional probability, we propose an algorithm
called \textit{class probability matching using kernel methods}
(\textit{CPMKM}) for label shift adaptation. From the theoretical perspective,
we establish the optimal convergence rates of CPMKM with respect to the
cross-entropy loss for multi-class label shift adaptation. From the
experimental perspective, comparisons on real datasets demonstrate that CPMKM
outperforms existing FPM-based and maximum-likelihood-based algorithms.

在领域适应中，协变量偏移和标签偏移问题是两个不同而互补的任务。针对标签偏移适应，提出了一种称为类概率匹配的新框架，通过匹配两个类概率函数在一维标签空间上估计类概率比率，与在 $d$ 维特征空间上操作的特征概率匹配方法有根本区别。将核逻辑回归纳入类概率匹配框架中，提出了一种称为基于核方法的类概率匹配的算法，用于标签偏移适应。从理论角度，建立了多类标签偏移适应中 CPMKM 相对于交叉熵损失的最优收敛速度。从实验角度，对真实数据集进行比较表明，CPMKM 优于现有基于 FPM 和最大似然的算法。