Convolutional Dictionary Learning (CDL) has emerged as a powerful approach for signal representation by learning translation-invariant features through convolution operations. While existing CDL methods are predominantly designed and used for fully supervised settings, many real-world classification tasks often rely on weakly labeled data, where only bag-level annotations are available. In this paper, we propose a novel weakly supervised convolutional dictionary learning framework that jointly learns shared and class-specific components, for multi-instance multi-label (MIML) classification where each example consists of multiple instances and may be associated with multiple labels. Our approach decomposes signals into background patterns captured by a shared dictionary and discriminative features encoded in class-specific dictionaries, with nuclear norm constraints preventing feature dilution. A Block Proximal Gradient method with Majorization (BPG-M) is developed to alternately update dictionary atoms and sparse coefficients, ensuring convergence to local minima. Furthermore, we incorporate a projection mechanism that aggregates instance-level predictions to bag-level labels through learnable pooling operators.Experimental results on both synthetic and real-world datasets demonstrate that our framework outperforms existing MIML methods in terms of classification performance, particularly in low-label regimes. The learned dictionaries provide interpretable representations while effectively handling background noise and variable-length instances, making the method suitable for applications such as environmental sound classification and RF signal analysis.