Recently, attention has been paid to tracking methods using sparse representation. Assuming that the representation residuals follow Gaussian distribution, the multi-object tracking methods based on sparse representation are proposed. However, these methods are sensitive to outliers such as occlusion due to the assumption of Gaussian distribution. In our paper, a novel sparse representation based multi-object tracking method is proposed via a tracking-by-detection scheme. Firstly, we find that the representation residuals of different occlusion instances follow the Laplacian distribution. Secondly, after the detection of the objects, a model named least absolute deviation with L1 regularization is proposed and applied to sparse representation of objects. The sparse solution of least absolute deviation problem is obtained by linear programming. Thirdly, an approach is proposed for discriminating the class of the detected objects. Meanwhile, an sparsity concentration index is introduced to distinguish new entered objects from existing objects. Experiments demonstrate that our method performs better than the state-of-the-art methods in persistent identity tracking.