Compressed sensing and hyper-reduction methods such as the Discrete Empirical Interpolation Method (DEIM) \cite{chaturantabut2010nonlinear} or Gappy POD (GPOD) \cite{everson1995karhunen} are widely used to approximate functions using limited information of it, and accelerate the solution of nonlinear partial differential equations (PDEs) within reduced-order models by approximating nonlinear operators through a small set of sampling points. While effective, classical methods rely on linear subspaces and greedy point selection, which can limit accuracy and robustness when approximating complex fields, especially when dealing with strongly nonlinear dynamics or varying problem configurations. In this work, we propose a novel nonlinear hyper-reduction framework based on the Rank Reduction AutoEncoder (RRAE) \cite{mounayer2024rank} combined with a learned masking strategy. The autoencoder learns a nonlinear low-dimensional representation of the data set, while an explicit truncated Singular Value Decomposition (SVD) layer in the latent space enforces low-rank structure. This acts as an intrinsic regularization, retaining only the most informative features and mitigating overfitting. Additionally, a trainable mask applied to the input data of the RRAE selects only the most informative points in order to ensure accurate field reconstruction, endowing the RRAE with compressed sensing capabilities. The resulting method can be interpreted as an efficient, nonlinear compressed sensing strategy, generalizing the principles of DEIM or GPOD to nonlinear manifolds, where now a linear model reduction exist but at the latent space of the RRAE and not in the original space where linear model-order reduction techniques fails to correctly approximate the data. In the context of nonlinear solvers, the proposed method provides an adaptive sampling strategy that improves approximation of Residual and Jacobian terms with fewer sampling points, enabling faster and more accurate reduced-order solutions of nonlinear PDEs. Numerical experiments demonstrate that the RRAE-based approach achieves higher accuracy and robustness than standard POD-DEIM solvers, while requiring fewer sampled nodes. Its ability to generalize to unseen parameters and loading conditions makes it particularly suitable for real-time hyper-reduction of large-scale nonlinear systems.