A novel low-complexity robust-to-noise iterative algorithm named compression-based gradient descent (C-GD) algorithm is proposed. C-GD is a generic compressed sensing recovery algorithm, that at its core, employs compression codes, such as JPEG2000 and MPEG4. Through using compression codes, C-GD strongly generalizes the scope of structures used by compressed sensing recovery algorithms beyond sparsity or low-rankness. The squared error of the proposed method and its associated convergence is characterized and predicts the strong performance of C-GD. Numerical results suggest that C-GD, when combined with state-of-the-art compression codes, either outperforms or performs comparably to modern compressed sensing recovery methods.