The multiple-instance learning (MIL) model has been successful in areas such as drug discovery and content-based image-retrieval. Recently, this model was generalized and a corresponding kernel was introduced to learn generalized MIL concepts with a support vector machine. While this kernel enjoyed empirical success, it has limitations in its representation. We extend this kernel by enriching its representation and empirically evaluate our new kernel on data from content-based image retrieval, biological sequence analysis, and drug discovery. We found that our new kernel generalized noticeably better than the old one in content-based image retrieval and biological sequence analysis and was slightly better or even with the old kernel in the other applications, showing that an SVM using this kernel does not overfit despite its richer representation.