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Non-negative matrix factorization (NMF) aims at finding non-negative representations of nonnegative data. Among different NMF algorithms, alternating direction method of multipliers (ADMM) is a popular one with superior performance. However, we find that ADMM shows instability and inferior performance on real-world data like speech signals. In this paper, to solve this problem, we develop a class of advanced regularized ADMM algorithms for NMF. Efficient and robust learning rules are achieved by incorporating l1-norm and the Frobenius norm regularization. The prior information of Laplacian distribution of data is used to solve the problem with a unique solution. We evaluate this class of ADMM algorithms using both synthetic and real speech signals for a source separation task at different cost functions, i.e., Euclidean distance (EUD), Kullback-Leibler (KL) divergence and Itakura-Saito (IS) divergence. Results demonstrate that the proposed algorithms converge faster and yield more stable and accurate results than the original ADMM algorithm.