Most of the existing dictionary learning models are based on linearly learned dictionaries, which have weak performance in nonlinear signal representation, thus driving a research boom in nonlinear dictionary learning (NLDL). In this paper, we propose a deep nonlinear dictionary learning model for dictionaries and coefficients with full-layer sparse regularizations to access deep latent information. We apply \(\ell _1\) regularization on the model to improve the efficiency of extracting key features hierarchically. We investigated the proposed algorithm using the Lifted Proximal Operator Machine (LPOM), by which a nonlinear model is transformed into a linear convex optimization problem to be solved. Then Nesterov’s acceleration is introduced to speed up the convergence, called \(\text {Accelerated DNLDL}\_\ell _1\) . We verify the feasibility of the proposed algorithm through numerical and application experiments. The results show that the acceleration scheme improves the convergence speed of the algorithm, and the proposed method has excellent performance on image classification and image denoising tasks.