In the last decade, seismic wavefield processing has begun to rely more heavily on the solution of wave-equation-based inverse problems. Especially when dealing with unfavourable data acquisition conditions (e.g., poor, regular or irregular sampling of sources and/or receivers), the underlying inverse problem is generally very ill-posed; sparsity promoting inversion coupled with fixed-basis sparsifying transforms has become the de-facto approach for many processing algorithms. Motivated by the ability of deep neural networks to identify compact representations of N-dimensional vector spaces, we propose to learn a mapping between the input seismic data and a latent manifold by means of an Autoencoder. The trained decoder is subsequently used as a nonlinear preconditioner for the inverse problem we wish to solve. Using joint deghosting and data reconstruction as an example, we show that nonlinear learned transforms outperform fixed-basis transforms and enable faster convergence to the sought solution (i.e, fewer applications of the forward and adjoint operators are required).