The wavefront sensor plays an important role in the adaptive optics (AO) system for aero-optical distortion correction. However, the bandwidth of the current data interfaces of wavefront sensors, as one of the key factors, limits applications of the AO system in extremely high-frequency aero-optical distortion correction, leading to unsatisfactory performance. In this paper, a framework for wavefront data compression using compressed sensing is established to improve the correction ability of the AO system, and a disturbed Zernike gradient dictionary (DZGD) learning over the k-singular value decomposition algorithm is proposed for achieving good performance in the compression of aero-optical wavefront data. Based on the proposed DZGD, a method for aero-optical distortion data compression and wavefront reconstruction is developed that can efficiently reduce the amount of data in the information channel without degradation of the correction effect in aero-optical distortion correction. The compressibility of aero-optical distortions over the DZGD is analyzed in detail by numerical simulations. In addition, the selection criteria of the measurement matrix and the anti-noise characteristic of the method are also discussed. Data compression using our method is feasible and highly adaptable in the correction of aero-optical distortions, and exhibits stronger resistance against detector noise compared with using the conventional dictionary.