Measurement and evaluation of multi-parametric CEST protocols requires complex and time consuming processing for correction of field inhomogeneities and contrast generation. In this work, we expand the linear projection approach for mapping motion corrected 7T CEST data directly to Lorentzian target parameters by L1-regularisation. This translates to subsampling in the frequency offset domain, resulting in reduced acquisition time. The method generalizes from healthy subject training data to unseen healthy test data and a tumor patient dataset. The L1-regularized linear projection approach integrates shortcutting of B0 and B1 correction, denoising, and Lorentzian fitting. It enforces sparsity of required frequency offsets.