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Matrix inversion problems are often encountered in experimental physics, andin particular in high-energy particle physics, under the name of unfolding. Thetrue 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 observedspectra into histograms, we can model the detector response via a matrix.Inferring a true spectrum starting from an observed spectrum requires thereforeinverting the response matrix. Many methods exist in literature for this task,all starting from the observed spectrum and using a simulated true spectrum asa guide to obtain a meaningful solution in cases where the response matrix isnot easily invertible.  In this Manuscript, I take a different approach to the unfolding problem.Rather than inverting the response matrix and transforming the observeddistribution into the most likely parent distribution in generator space, Isample many distributions in generator space, fold them through the originalresponse matrix, and pick the generator-level distribution that yields thefolded distribution closest to the data distribution. Regularization schemescan be introduced to treat the case where non-diagonal response matrices resultin high-frequency oscillations of the solution in true space, and theintroduced bias is studied.  The algorithm performs as well as traditional unfolding algorithms in caseswhere the inverse problem is well-defined in terms of the discretization of thetrue and smeared space, and outperforms them in cases where the inverse problemis ill-defined---when the number of truth-space bins is larger than that ofsmeared-space bins. These advantages stem from the fact that the algorithm doesnot technically invert any matrix and uses only the data distribution as aguide to choose the best solution.

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