We provide two compressive sensing (CS) recovery algorithms based on iterative hard-thresholding. The algorithms, collectively dubbed as algebraic pursuits (ALPS), exploit the restricted isometry properties of the CS measurement matrix within the algebra of Nesterov's optimal gradient methods. We theoretically characterize the approximation guarantees of ALPS for signals that are sparse on ortho-bases as well as on tight-frames. Simulation results demonstrate a great potential for ALPS in terms of phase-transition, noise robustness, and CS reconstruction.