Context. Inversion codes are numerical tools used to infer physical properties from observations. Despite their success, the quality of current spectropolarimetric observations and those expected in the near future presents a challenge to current inversion codes. Aims: The pixel-by-pixel strategy of inverting spectropolarimetric data that we currently use needs to be surpassed and improved. The inverted physical parameters have to take into account the spatial correlation that is present in the data and that contains valuable physical information. Methods: We used the concept of sparsity or compressibility to develop a new generation of inversion codes for the Stokes parameters. The inversion code uses numerical optimization techniques based on the idea of proximal algorithms to impose sparsity. In so doing, we allow for the first time exploiting the spatial correlation on the maps of physical parameters. Sparsity also regularizes the solution by reducing the number of unknowns. Results: We compare the results of the new inversion code with pixel-by-pixel inversions to demonstrate the increased robustness of the solution. We also show how the method can easily compensate for the effect of the telescope point spread function, producing solutions with an enhanced contrast.