The paper presents a general formulation for supervised dictionary learning, which is adapted to a wide variety of tasks such as classification, regression, and compressed sensing. The authors propose an efficient algorithm to solve the corresponding optimization problem, demonstrating its effectiveness through experiments on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing. The main contributions of the paper include: 1. **Supervised Formulation**: Introduces a supervised formulation for learning dictionaries adapted to specific tasks, in contrast to unsupervised formulations used for data reconstruction. 2. **Smooth Optimization Problem**: Shows that the resulting optimization problem is smooth under mild assumptions, and empirically demonstrates that stochastic gradient descent addresses it efficiently. 3. **Semi-Supervised Learning**: Demonstrates that the proposed formulation is well-suited for semi-supervised learning, leveraging unlabeled data when it admits sparse representations, and achieving state-of-the-art results in various machine learning and signal processing problems. The paper also discusses extensions of the formulation, including the inclusion of a linear transform of the input data and the exploitation of unlabeled data in semi-supervised settings. Experimental results on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing validate the effectiveness of the proposed approach.The paper presents a general formulation for supervised dictionary learning, which is adapted to a wide variety of tasks such as classification, regression, and compressed sensing. The authors propose an efficient algorithm to solve the corresponding optimization problem, demonstrating its effectiveness through experiments on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing. The main contributions of the paper include: 1. **Supervised Formulation**: Introduces a supervised formulation for learning dictionaries adapted to specific tasks, in contrast to unsupervised formulations used for data reconstruction. 2. **Smooth Optimization Problem**: Shows that the resulting optimization problem is smooth under mild assumptions, and empirically demonstrates that stochastic gradient descent addresses it efficiently. 3. **Semi-Supervised Learning**: Demonstrates that the proposed formulation is well-suited for semi-supervised learning, leveraging unlabeled data when it admits sparse representations, and achieving state-of-the-art results in various machine learning and signal processing problems. The paper also discusses extensions of the formulation, including the inclusion of a linear transform of the input data and the exploitation of unlabeled data in semi-supervised settings. Experimental results on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing validate the effectiveness of the proposed approach.