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This work aims to address two issues that often exist in least square regression (LSR) models for classification tasks, which are (1) learning a compact projection matrix for feature selection and (2) adopting relaxed regression targets. To this end, we first propose a sparse regularized LSR framework for feature selection by introducing the ℓ 2 , 1 regularizer. Second, we utilize two different strategies to relax the strict regression targets based on the sparse framework. One way is to exploit the -dragging technique. Another strategy is to directly learn the labels from the inputs and constrain the distance between true and false classes simultaneously. Hence, more feasible regression schemes are constructed, and the models will be more flexible. Further, efficient iterative methods are derived to optimize the proposed models. Various experiments on image databases intend to manifest our proposed models have outstanding recognition capability compared with many state-of-the-art classifiers. [ABSTRACT FROM AUTHOR]