As to detect and correct the labeling errors, a labeling errors detecting and correcting algorithm based on sparse manifold clustering and embedding and L₁norm regularization is proposed. The proposed algorithm is based on the useful information in the original labels and the natural discriminating information which is contained in the data structure. Firstly, the original data are projected to the new space by using the sparse manifold clustering and embedding model. Then, a nearest neighbor classifier and a very small amount samples which are labeled correctly are used to obtain new labels for the original data. Meanwhile, the constructing labeling error detection model is built and then the sparse label detection vector which consists of 0 and 1 is obtained to modify the detection errors. The inaccurate and accurate labels correspond to 0 and 1 in the label detection vector respectively. Finally, the convex optimization scheme is introduced to solve the optimization problem and the convergence proofs are given. The experiment results show the effectiveness of the proposed algorithm based on the artificial data of complex manifold structure and the typical low-dimensional, high-dimensional data.