Incomplete features and label noise in multi-view multi-label data significantly undermine the reliability and performance, motivating researchers to explore the mechanism of representation and information recovery. However, learning for such dual deficiencies is crucial but rarely studied. In this paper, we propose a theory-inspired Deep Multi-View Multi-Label Learning method with Incomplete Views and Noisy Labels named DMMIvNL to address these problems.Specifically, to promote the synthesis of task-relevant shared information and preserve the distinctiveness of individual features from limited views, we have developed a feature extraction modular based on the information bottleneck theory, and formulated its theoretical upper bound into its objective.Meanwhile, we theoretically prove that minimizing the volume of the transition matrix ensures the statistical consistency with classifier training. Besides, a cycle-consistent estimation principle is proposed in the volume minimization network to improve the recognition stability of multi-label noise. Moreover, leveraging inherent real semantics information and label correlations are employed as model regularization to reduce the risk of excessive noise fitting.Finally, extensive experimental results validate the effectiveness and robustness of our DMMIvNL.