论文标题
通过折叠进行展开：对矩阵反转问题的重新采样方法而不实际反转任何矩阵
Unfolding by Folding: a resampling approach to the problem of matrix inversion without actually inverting any matrix
论文作者
论文摘要
矩阵反转问题通常以实验物理学，尤其是在高能粒子物理学中遇到，以展开的名义遇到。物理量的真实光谱因探测器的存在而变形，从而导致观察到的光谱。如果我们将真实光谱和观察到的光谱分散到直方图中，则可以通过矩阵对检测器响应进行建模。从观察到的频谱开始，推断出真正的频谱需要反转响应矩阵。文献中存在许多方法的此任务，所有方法都是从观察到的频谱开始的，然后使用模拟的真实频谱作为指南，以在响应矩阵不易可逆的情况下获得有意义的解决方案。 在此手稿中，我对发展的问题采取了另一种方法。我没有反转响应矩阵并将观察到的分布转换为发电机空间中最可能的母体分布，而是在发电机空间中采样了许多分布，而是通过原始响应矩阵折叠它们，然后选择产生最接近数据分布的折叠分布的生成器级分布。可以引入正则化方案来治疗非对角响应矩阵导致溶液在真实空间中的高频振荡并研究了引入的偏见的情况。 在逆问题在真实和污迹空间的离散化方面明确定义的情况下，该算法的性能以及传统展开算法的性能，并且在逆问题不确定的情况下 - 当真相空间箱的数量大于碰撞空间箱的数量大于Smeared Space Bins的情况。这些优点源于以下事实：算法在技术上没有扭转任何矩阵，而仅使用数据分发作为选择最佳解决方案的指南。
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.
