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Predictive risk estimation for the expectation maximization algorithm with Poisson data
Inverse Problems ( IF 2.1 ) Pub Date : 2021-03-18 , DOI: 10.1088/1361-6420/abe950
Paolo Massa , Federico Benvenuto

In this work, we introduce a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback–Leibler divergence, in order to define a regularization parameter’s choice rule for the expectation maximization (EM) algorithm. To this aim, we prove a Poisson counterpart of the Stein’s Lemma for Gaussian variables, and from this result we derive the proposed estimator showing its analogies with the well-known Stein’s unbiased risk estimator valid for a quadratic loss. We prove that the proposed estimator is asymptotically unbiased with increasing number of measured counts, under certain mild conditions on the regularization method. We show that these conditions are satisfied by the EM algorithm under the hypothesis that the underlying matrix has positive entries and then we apply this estimator to select the EM optimal reconstruction. We present some numerical tests in the case of image deconvolution, comparing the performances of the proposed estimator with other methods available in the literature, both in the inverse crime and non-inverse crime setting.



中文翻译:

具有泊松数据的期望最大化算法的预测风险估计

在这项工作中,当损失函数是 Kullback-Leibler 散度时,我们引入了一种新的泊松数据预测风险估计器,以便为期望最大化 (EM) 算法定义正则化参数的选择规则。为此,我们证明了高斯变量的 Stein 引理的 Poisson 对应物,并且从这个结果我们推导出了建议的估计量,显示其与众所周知的 Stein 无偏风险估计量的类比,该估计量对二次损失有效。我们证明,在正则化方法的某些温和条件下,所提出的估计量随着测量计数数量的增加而渐近无偏。我们表明,在基础矩阵具有正项的假设下,EM 算法满足这些条件,然后我们应用此估计器来选择 EM 最佳重建。我们在图像去卷积的情况下提供了一些数值测试,将所提出的估计器的性能与文献中可用的其他方法进行了比较,无论是在逆犯罪和非逆犯罪环境中。

更新日期:2021-03-18
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