Fredholm积分反演的投影回归方法：形式化及其在解析延拓中的应用
Projected Regression Methods for Inverting Fredholm Integrals: Formalism and Application to Analytical Continuation
摘要：
我们提出了一种机器学习方法的第一类Fredholm积分的反演。该方法在Fredholm核的逆是病态的情况下提供了自然的正则化。它还提供了一种有效且稳定的约束处理方法。关键的观察是，稳定的前向问题允许建设一个大型数据库的输出物理上有意义的投入。我们将机器学习应用于该数据库，以生成具有受控复杂性的回归函数，该函数返回先前未见过的输入的近似解;然后将近似解投影到满足相关约束的函数的子空间上。我们还推导并提出了不确定性估计。我们说明的方法，将其应用到量子多体物理学的解析延拓问题，其中涉及重建的物理激发光谱的频率依赖性从在复杂的频率平面中的特定点获得的数据。在标准误差度量下，对于低输入噪声，该方法的性能与最大熵方法一样好或更好，并且对增加的输入噪声具有显著更强的鲁棒性。我们期望的方法是类似的有效的任何问题，涉及正式病态反演，只要前向问题可以有效地解决。
英文摘要：
We present a machine learning approach to the inversion of Fredholm integrals of the first kind. The approach provides a natural regularization in cases where the inverse of the Fredholm kernel is ill-conditioned. It also provides an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. We apply machine learning to this database to generate a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. We also derive and present uncertainty estimates. We illustrate the approach by applying it to the analytical continuation problem of quantum many-body physics, which involves reconstructing the frequency dependence of physical excitation spectra from data obtained at specific points in the complex frequency plane. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We expect the methodology to be similarly effective for any problem involving a formally ill-conditioned inversion, provided that the forward problem can be efficiently solved.
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