Feature selection methods are widely used in machine learning tasks to reduce the dimensionality and improve the performance of the models. However, traditional feature selection methods based on regression often suffer from a lack of robustness and generalization ability and are easily affected by outliers in the data. To address this problem, we propose a robust feature selection method based on sparse regression. This method uses a non-square form of the L2,1 norm as both the loss function and regularization term, which can effectively enhance the model’s resistance to outliers and achieve feature selection simultaneously. Furthermore, to improve the model’s robustness and prevent overfitting, we add an elastic variable to the loss function. We design two efficient convergent iterative processes to solve the optimization problem based on the L2,1 norm and propose a robust joint sparse regression algorithm. Extensive experimental results on three public datasets show that our feature selection method outperforms other comparison methods.