Statistical-mechanical analysis of dynamical properties of algorithms for image restoration

在本研究项目中，我们研究了分别由Ising模型和Q-Ising模型表示的二值图像和灰度图像的贝叶斯图像复原的动力学性质。我们引入平均场模型来评估EM算法的收敛精度或收敛速度，并引入梯度下降法来获得超参数的最大似然估计。结果表明，EM算法在收敛速度上优于梯度下降法。此外，我们的分析表明，当等待系统达到平衡态的时间不够时，超参数的动力学行为在它们收敛到真值之前表现出一种“振荡现象”。我们还提出了一种新型的确定性退火法EM算法，并利用非广泛的统计力学对其进行了推广。我们还给出了量子自旋玻璃(随机场伊辛模型、Shermgton-Kjrkpatrick模型、横场中存在p体相互作用的伊辛自旋玻璃)在概率信息处理中的几个应用，特别是在图像恢复和纠错编码问题上。作为一种相关的优化方法，量子退火法也被引入到这些研究领域，并用量子马尔科夫链蒙特卡罗方法对其性能进行了研究。导出了获得最佳量子MPM估计性能的Nishimori-Wong条件作为横场有效振幅的条件。

In this research project, we investigated the dynamical properties of Bayesian image restoration for the blapk-while binary and gray-scaled images represented by the Ising and the Q-Ising models, respectively. We introduced the mean-field model to evaluate the accuracy or the speed of the convergence for EM algorithm and gradient descent method to obtain the maximum likelihood estimate for the hyper-parameters. We found that EM algorithm is superior to the gradient descent method in the speed of the convergence. Moreover, our analysis made it clear that the dynamical behavior of the hyper-parameters shows a kind of the "oscillating phenomena " before they converge to their true values if the time to wait until the system reaches to the equilibrium states is not enough.We also proposed a new-type of the deterministic annealing EM algorithm which was extended by means of non-extensive statistical mechanics. The method was applied to the problem of Traveling salesman and the statistical performance was investigated extensively.We also presented several applications of quantum spin glasses (random field Ising model, Shermgton-KJrkpatrick model, Ising spin glasses with p-body interaction in a transverse field) to probabilistic information processing, especially to the problems of image restoration and error-correcting codes. As a related optimization method, quantum annealing was also introduced to these research fields and its performance was investigated by using the quantum Markov chain Monte Carlo method. The Nishimori-Wong condition, on which the best possible performance of the quantum MPM estimation is achieved, was derived as a condition on the effective amplitude of the transverse field.