This paper provides a comprehensive survey of sparse representation methods, their algorithms, and applications. Sparse representation has gained significant attention in signal processing, image processing, computer vision, and pattern recognition due to its effectiveness in both theory and practice. The paper categorizes sparse representation methods based on different norm minimizations, including $ l_0 $-norm, $ l_p $-norm (0 < p < 1), $ l_1 $-norm, and $ l_{2,1} $-norm. It also classifies algorithms into four groups: greedy strategy approximation, constrained optimization, proximity algorithm-based optimization, and homotopy algorithm-based sparse representation. The paper analyzes the rationale of these algorithms and summarizes a wide range of applications, demonstrating the potential of sparse representation theory. Sparse representation is closely related to compressed sensing (CS), which allows for the reconstruction of signals from a small number of measurements. CS theory includes three components: sparse representation, encoding measurement, and reconstruction algorithm. Sparse representation is essential for overcoming the limitations of traditional sampling methods like Shannon's sampling theorem. It has been successfully applied to various tasks, including image classification, signal processing, and data reconstruction. The paper discusses different approaches to sparse representation, including greedy algorithms like Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP), constrained optimization methods, and proximity algorithms. It also covers the $ l_1 $-norm minimization approach, which is computationally efficient and has been widely used in sparse representation. The paper provides a detailed analysis of these methods, their mathematical formulations, and their applications in various fields. The paper concludes that sparse representation is a powerful tool with broad applications in signal processing, image processing, and machine learning. It emphasizes the importance of understanding the theoretical foundations and practical implementations of sparse representation methods to advance research and applications in these areas.This paper provides a comprehensive survey of sparse representation methods, their algorithms, and applications. Sparse representation has gained significant attention in signal processing, image processing, computer vision, and pattern recognition due to its effectiveness in both theory and practice. The paper categorizes sparse representation methods based on different norm minimizations, including $ l_0 $-norm, $ l_p $-norm (0 < p < 1), $ l_1 $-norm, and $ l_{2,1} $-norm. It also classifies algorithms into four groups: greedy strategy approximation, constrained optimization, proximity algorithm-based optimization, and homotopy algorithm-based sparse representation. The paper analyzes the rationale of these algorithms and summarizes a wide range of applications, demonstrating the potential of sparse representation theory. Sparse representation is closely related to compressed sensing (CS), which allows for the reconstruction of signals from a small number of measurements. CS theory includes three components: sparse representation, encoding measurement, and reconstruction algorithm. Sparse representation is essential for overcoming the limitations of traditional sampling methods like Shannon's sampling theorem. It has been successfully applied to various tasks, including image classification, signal processing, and data reconstruction. The paper discusses different approaches to sparse representation, including greedy algorithms like Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP), constrained optimization methods, and proximity algorithms. It also covers the $ l_1 $-norm minimization approach, which is computationally efficient and has been widely used in sparse representation. The paper provides a detailed analysis of these methods, their mathematical formulations, and their applications in various fields. The paper concludes that sparse representation is a powerful tool with broad applications in signal processing, image processing, and machine learning. It emphasizes the importance of understanding the theoretical foundations and practical implementations of sparse representation methods to advance research and applications in these areas.