针对超出最小二乘的可分离逆问题的变量消去推广
A generalization of variable elimination for separable inverse problems beyond least squares
AI总结：
针对现有变量消去方法局限于最小二乘的缺陷，提出一种通用框架，通过修改标准优化方法模拟变量消去，适用于泊松似然、边界约束等问题，并在指数和拟合与盲反卷积测试中展现出速度和鲁棒性优势。
摘要：
在线性逆问题中，我们拥有从某些未知参数的含噪线性变换得到的数据，并希望从数据中估计这些未知量。可分离逆问题是一种强大的推广，其中变换本身依赖于额外的未知参数，我们希望同时确定两组参数。当通过优化求解可分离问题时，通常可以通过消去线性变量来加速收敛，这一策略最突出地出现在Golub、Pereyra和Kaufman的变量投影方法中。现有的变量消去方法需要线性变量最优值的显式公式，因此无法用于具有泊松似然、边界约束或其他重要偏离最小二乘的问题。为解决这一限制，我们提出了一种变量消去的推广，其中标准优化方法被修改为表现得好像变量已被消去。我们通过使用该方法重新推导几种现有的变量消去技术，验证了该方法是适当的推广。然后，我们将该方法扩展到边界约束和泊松问题，在此过程中表明，变量消去方法的许多最佳特征可以在我们的框架中复制。在困难的指数和拟合与盲反卷积问题上的测试表明，所提出的方法在速度和鲁棒性方面相对于标准方法具有显著优势。
英文摘要：
In linear inverse problems, we have data derived from a noisy linear transformation of some unknown parameters, and we wish to estimate these unknowns from the data. Separable inverse problems are a powerful generalization in which the transformation itself depends on additional unknown parameters and we wish to determine both sets of parameters simultaneously. When separable problems are solved by optimization, convergence can often be accelerated by elimination of the linear variables, a strategy which appears most prominently in the variable projection methods due to Golub, Pereyra, and Kaufman. Existing variable elimination methods require an explicit formula for the optimal value of the linear variables, so they cannot be used in problems with Poisson likelihoods, bound constraints, or other important departures from least squares. To address this limitation, we propose a generalization of variable elimination in which standard optimization methods are modified to behave as though a variable has been eliminated. We verify that this approach is a proper generalization by using it to re-derive several existing variable elimination techniques. We then extend the approach to bound-constrained and Poissonian problems, showing in the process that many of the best features of variable elimination methods can be duplicated in our framework. Tests on difficult exponential sum fitting and blind deconvolution problems indicate that the proposed approach can have significant speed and robustness advantages over standard methods.
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