This paper presents a theoretical analysis of the iterative hard thresholding (IHT) algorithm for compressed sensing. The algorithm is shown to have several desirable properties: it provides near-optimal error guarantees, is robust to observation noise, requires a minimum number of observations, and can be used with any sampling operator for which the operator and its adjoint can be computed. The algorithm has linear memory requirements and computational complexity per iteration comparable to the application of the measurement operator or its adjoint. It requires a fixed number of iterations depending on the logarithm of a signal-to-noise ratio and has uniform performance guarantees that depend only on the properties of the sampling operator and signal sparsity. The paper discusses the theoretical foundations of compressed sensing, including the restricted isometry property (RIP), which is crucial for the algorithm's performance. It also compares the IHT algorithm to other methods like CoSaMP, showing that IHT has similar performance guarantees but with a simpler implementation. The paper proves that IHT can recover signals with high accuracy, and it provides bounds on the error in terms of the signal's sparsity and the observation noise. The algorithm is shown to require a logarithmic number of iterations relative to the signal-to-noise ratio, making it efficient for practical applications. The paper also discusses the implications of these results for signal recovery and the limitations of uniform performance guarantees in practical scenarios.This paper presents a theoretical analysis of the iterative hard thresholding (IHT) algorithm for compressed sensing. The algorithm is shown to have several desirable properties: it provides near-optimal error guarantees, is robust to observation noise, requires a minimum number of observations, and can be used with any sampling operator for which the operator and its adjoint can be computed. The algorithm has linear memory requirements and computational complexity per iteration comparable to the application of the measurement operator or its adjoint. It requires a fixed number of iterations depending on the logarithm of a signal-to-noise ratio and has uniform performance guarantees that depend only on the properties of the sampling operator and signal sparsity. The paper discusses the theoretical foundations of compressed sensing, including the restricted isometry property (RIP), which is crucial for the algorithm's performance. It also compares the IHT algorithm to other methods like CoSaMP, showing that IHT has similar performance guarantees but with a simpler implementation. The paper proves that IHT can recover signals with high accuracy, and it provides bounds on the error in terms of the signal's sparsity and the observation noise. The algorithm is shown to require a logarithmic number of iterations relative to the signal-to-noise ratio, making it efficient for practical applications. The paper also discusses the implications of these results for signal recovery and the limitations of uniform performance guarantees in practical scenarios.