This paper presents two iterative algorithms for sparse approximation: the iterative hard-thresholding algorithm for the $ \ell_0 $-regularized problem and the M-sparse algorithm for the M-sparse constrained problem. Both algorithms aim to minimize the cost functions associated with sparse signal approximations. The $ \ell_0 $-regularized problem involves minimizing a cost function that includes an $ \ell_0 $-norm penalty, while the M-sparse problem involves minimizing a cost function with a constraint on the number of non-zero coefficients. The iterative hard-thresholding algorithm is derived by applying a hard thresholding operator to the result of a gradient descent step. This algorithm is guaranteed to not increase the cost function and converges to a local minimum. The M-sparse algorithm similarly uses a thresholding operator to retain only the M largest coefficients, and is also guaranteed to not increase the cost function and converge to a local minimum. Both algorithms are shown to be effective in improving the performance of other methods such as Matching Pursuit and Orthogonal Matching Pursuit. The M-sparse algorithm can also be used independently and is shown to perform well in certain cases. The algorithms are computationally efficient, with each iteration having a complexity similar to that of Matching Pursuit. The paper also provides theoretical results showing that the algorithms have multiple fixed points and that their performance depends on the properties of the dictionary used. Numerical studies demonstrate that the algorithms can achieve better results than other methods in certain scenarios.This paper presents two iterative algorithms for sparse approximation: the iterative hard-thresholding algorithm for the $ \ell_0 $-regularized problem and the M-sparse algorithm for the M-sparse constrained problem. Both algorithms aim to minimize the cost functions associated with sparse signal approximations. The $ \ell_0 $-regularized problem involves minimizing a cost function that includes an $ \ell_0 $-norm penalty, while the M-sparse problem involves minimizing a cost function with a constraint on the number of non-zero coefficients. The iterative hard-thresholding algorithm is derived by applying a hard thresholding operator to the result of a gradient descent step. This algorithm is guaranteed to not increase the cost function and converges to a local minimum. The M-sparse algorithm similarly uses a thresholding operator to retain only the M largest coefficients, and is also guaranteed to not increase the cost function and converge to a local minimum. Both algorithms are shown to be effective in improving the performance of other methods such as Matching Pursuit and Orthogonal Matching Pursuit. The M-sparse algorithm can also be used independently and is shown to perform well in certain cases. The algorithms are computationally efficient, with each iteration having a complexity similar to that of Matching Pursuit. The paper also provides theoretical results showing that the algorithms have multiple fixed points and that their performance depends on the properties of the dictionary used. Numerical studies demonstrate that the algorithms can achieve better results than other methods in certain scenarios.