Non-Convex Sparse Deviation Modeling Via Generative Models

In this paper, the generative model is used to introduce the structural properties of the signal to replace the common sparse hypothesis, and a non-convex compressed sensing sparse deviation model based on the generative model ($\ell_q$-Gen) is proposed. By establishing $\ell_q$ variant of the restricted isometry property ($q$-RIP) and Set-Restricted Eigenvalue Condition ($q$-$S$-REC), the error upper bound of the optimal decoder is derived when the recovered signal is within the sparse deviation range of the generator. Furthermore, it is proved that the Gaussian matrix satisfying a certain number of measurements is sufficient to ensure a good recovery for the generating function with high probability. Finally, a series of experiments are carried out to verify the effectiveness and superiority of the $\ell_q$-Gen model.