The impulse noise is difficult to deal with by using the existing algorithms. A WSN localization algorithm based on the Bregman divergence was proposed to improve the node positioning accuracy of the wireless sensor network (WSN). The proposed algorithm is divided into two stages, the stage of Euclidean distance matrix (EDM) recovery and the stage of coordinates mapping. The problem of EDM recovery is transformed to an issue of matrix completion in a noisy environment, based on the natural low-rank character of the EDM. A regularized matrix completion model is established by using L_1,2-norm explicit smoothing the pulse noise. To solve the model effectively, the multivariate function Bregman divergence was defined, then the split Bregman iteration was extended to the matrix space, and the alternating minimization method was combined to obtain the EDM estimator. Furthermore, node localization was available based on the multi-dimensional scaling method. Experiments demonstrate that under different noise conditions, the proposed algorithm outperforms other algorithms in terms of positioning accuracy and robustness while ensuring high efficiency. Notably, the localization error of the proposed algorithm was less than a quarter of those of other algorithms, when the sampling rate reached a certain level.