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As a promising solution of reducing annotation cost, training multi-label models with partial positive labels (MLR-PPL), in which merely few positive labels are known while other are missing, attracts increasing attention. Due to the absence of any negative labels, previous works regard unknown labels as negative and adopt traditional MLR algorithms. To reject noisy labels, recent works regard large loss samples as noise but ignore the semantic correlation different multi-label images. In this work, we propose to explore semantic correlation among different images to facilitate the MLR-PPL task. Specifically, we design a unified framework, Category-Adaptive Label Discovery and Noise Rejection, that discovers unknown labels and rejects noisy labels for each category in an adaptive manner. The framework consists of two complementary modules: (1) Category-Adaptive Label Discovery module first measures the semantic similarity between positive samples and then complement unknown labels with high similarities; (2) Category-Adaptive Noise Rejection module first computes the sample weights based on semantic similarities from different samples and then discards noisy labels with low weights. Besides, we propose a novel category-adaptive threshold updating that adaptively adjusts the threshold, to avoid the time-consuming manual tuning process. Extensive experiments demonstrate that our proposed method consistently outperforms current leading algorithms.