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We formulate a practical yet challenging problem: General Partial Label Learning (GPLL). Compared to the traditional Partial Label Learning (PLL) problem, GPLL relaxes the supervision assumption from instance-level -- a label set partially labels an instance -- to group-level: 1) a label set partially labels a group of instances, where the within-group instance-label link annotations are missing, and 2) cross-group links are allowed -- instances in a group may be partially linked to the label set from another group. Such ambiguous group-level supervision is more practical in real-world scenarios as additional annotation on the instance-level is no longer required, e.g., face-naming in videos where the group consists of faces in a frame, labeled by a name set in the corresponding caption. In this paper, we propose a novel graph convolutional network (GCN) called Dual Bipartite Graph Autoencoder (DB-GAE) to tackle the label ambiguity challenge of GPLL. First, we exploit the cross-group correlations to represent the instance groups as dual bipartite graphs: within-group and cross-group, which reciprocally complements each other to resolve the linking ambiguities. Second, we design a GCN autoencoder to encode and decode them, where the decodings are considered as the refined results. It is worth noting that DB-GAE is self-supervised and transductive, as it only uses the group-level supervision without a separate offline training stage. Extensive experiments on two real-world datasets demonstrate that DB-GAE significantly outperforms the best baseline over absolute 0.159 F1-score and 24.8% accuracy. We further offer analysis on various levels of label ambiguities.