Unfolding by Folding: a resampling approach to the problem of matrix inversion without actually inverting any matrix

Matrix inversion problems are often encountered in experimental physics, and in particular in high-energy particle physics, under the name of unfolding. The true spectrum of a physical quantity is deformed by the presence of a detector, resulting in an observed spectrum. If we discretize both the true and observed spectra into histograms, we can model the detector response via a matrix. Inferring a true spectrum starting from an observed spectrum requires therefore inverting the response matrix. Many methods exist in literature for this task, all starting from the observed spectrum and using a simulated true spectrum as a guide to obtain a meaningful solution in cases where the response matrix is not easily invertible. In this Manuscript, I take a different approach to the unfolding problem. Rather than inverting the response matrix and transforming the observed distribution into the most likely parent distribution in generator space, I sample many distributions in generator space, fold them through the original response matrix, and pick the generator-level distribution that yields the folded distribution closest to the data distribution. Regularization schemes can be introduced to treat the case where non-diagonal response matrices result in high-frequency oscillations of the solution in true space, and the introduced bias is studied. The algorithm performs as well as traditional unfolding algorithms in cases where the inverse problem is well-defined in terms of the discretization of the true and smeared space, and outperforms them in cases where the inverse problem is ill-defined---when the number of truth-space bins is larger than that of smeared-space bins. These advantages stem from the fact that the algorithm does not technically invert any matrix and uses only the data distribution as a guide to choose the best solution.