Multi-layer networks provide an effective and efficient tool to model and characterize complex systems with multiple types of interactions, which differ greatly from the traditional single-layer networks. Graph clustering in multi-layer networks is highly non-trivial since it is difficult to balance the connectivity of clusters and the connection of various layers. The current algorithms for the layer-specific clusters are criticized for the low accuracy and sensitivity to the perturbation of networks. To overcome these issues, a novel algorithm for the layer-specific module in multi-layer networks based on nonnegative matrix factorization (LSNMF) is proposed by explicitly exploring the specific features of vertices. LSNMF first extract features of vertices in multi-layer networks by using nonnegative matrix factorization (NMF) and then decompose features of vertices into the common and specific components. The orthogonality constraint is imposed on the specific components to ensure the specificity of features of vertices, which provides a better strategy to characterize and model the structure of layer-specific modules. The extensive experiments demonstrate that the proposed algorithm dramatically outperforms state-of-the-art baselines in terms of various measurements. Furthermore, LSNMF efficiently extracts stage-specific modules, which are more likely to enrich the known functions, and also associate with the survival time of patients.