Open-set classification is a common problem in many real world tasks, where data is collected for known classes, and some novel classes occur at the test stage. In this paper, we focus on a more challenging case where the data examples collected for known classes are all unlabeled. Due to the high cost of label annotation, it is rather important to train a model with least labeled data for both accurate classification on known classes and effective detection of novel classes. Firstly, we propose an active learning method by incorporating structured sparsity with diversity to select representative examples for annotation. Then a latent low-rank representation is employed to simultaneously perform classification and novel class detection. Also, the method along with a fast optimization solution is extended to a multi-stage scenario, where classes occur and disappear in batches at each stage. Experimental results on multiple datasets validate the superiority of the proposed method with regard to different performance measures.