This paper presents a theoretical analysis of the iterative hard thresholding (IHT) algorithm for compressed sensing. The authors show that IHT provides near-optimal error guarantees and can recover signals with high accuracy from compressed sensing observations. The main result states that if the observation matrix $\Phi$ satisfies the restricted isometry property with $\beta_{3s} < 1/8$, then IHT reduces the estimation error in each iteration and is guaranteed to achieve a constant factor of the best attainable estimation error. The number of iterations required to achieve a desired accuracy depends on the logarithm of the signal-to-noise ratio. The paper also discusses a stopping criterion for the algorithm and compares IHT to other state-of-the-art algorithms like CoSAMP, highlighting their performance differences in both theoretical and numerical studies. The authors conclude by summarizing the key properties of IHT, emphasizing its simplicity and effectiveness in compressed sensing applications.This paper presents a theoretical analysis of the iterative hard thresholding (IHT) algorithm for compressed sensing. The authors show that IHT provides near-optimal error guarantees and can recover signals with high accuracy from compressed sensing observations. The main result states that if the observation matrix $\Phi$ satisfies the restricted isometry property with $\beta_{3s} < 1/8$, then IHT reduces the estimation error in each iteration and is guaranteed to achieve a constant factor of the best attainable estimation error. The number of iterations required to achieve a desired accuracy depends on the logarithm of the signal-to-noise ratio. The paper also discusses a stopping criterion for the algorithm and compares IHT to other state-of-the-art algorithms like CoSAMP, highlighting their performance differences in both theoretical and numerical studies. The authors conclude by summarizing the key properties of IHT, emphasizing its simplicity and effectiveness in compressed sensing applications.