We propose a new method for robust learning Laplacian matrices from observed smooth graph signals in the presence of both Gaussian noise and random-valued impulse noise (i.e., outliers). Using the recently developed factor analysis model for representing smooth graph signals in [1], we formulate our learning process as a constrained optimization problem, and adopt the £i-norm for measuring the data fidelity in order to improve robustness. Computational results on three types of synthetic graphs demonstrate that the proposed method outperforms the state-of-the-art methods in terms of commonly used information retrieval metrics, such as F-measure, precision, recall and normalized mutual information. In particular, we observed that F-measure is improved by up to 16%.