In this work, we propose a novel neural network architecture, called Adaptive Convolutional Neural Network (ACNN), which can be viewed as an intermediate solution between a standard convolutional network and a fully connected one. A constrained training strategy is developed to learn the parameters of such a network. The proposed algorithm allows us to control the Lipschitz constant of our ACNN to secure its robustness to adversarial noise. The resulting learning approach is evaluated for signal denoising based on a database of music recordings. Both qualitative and quantitative results show that the designed network is successful in removing Gaussian noise with unknown variance