Context. The unprecedented amount and the excellent quality of lensing data expected from upcoming ground and space-based surveys present a great opportunity for shedding light on questions that remain unanswered with regard to our universe and the validity of the standard ΛCDM cosmological model. The development of new techniques that are capable of exploiting the vast quantity of data provided by future observations, in the most effective way possible, is of great importance.Aims. This is the reason we chose to investigate the development of a new method for treating weak-lensing higher-order statistics, which are known to break the degeneracy among cosmological parameters thanks to their capacity to probe non-Gaussian properties of the shear field. In particular, the proposed method applies directly to the observed quantity, namely, the noisy galaxy ellipticity.Methods. We produced simulated lensing maps with different sets of cosmological parameters and used them to measure higher-order moments, Minkowski functionals, Betti numbers, and other statistics related to graph theory. This allowed us to construct datasets with a range of sizes, levels of precision, and smoothing. We then applied several machine learning algorithms to determine which method best predicts the actual cosmological parameters associated with each simulation.Results. The most optimal model turned out to be a simple multidimensional linear regression. We use this model to compare the results coming from the different datasets and find that we can measure, with a good level of accuracy, the majority of the parameters considered in this study. We also investigated the relation between each higher-order estimator and the different cosmological parameters for several signal-to-noise thresholds and redshifts bins.Conclusions. Given the promising results we obtained, we consider this approach a valuable resource that is worthy of further development.Key words: gravitational lensing: weak / cosmology: theory / methods: statistical