Statistical mechanics of relaxation process for Bayesian inference and its application

我们研究了码分多址多用户检测的动力学问题。将基于高斯近似的统计神经动力学应用于并行干扰对消器(PIC)的检测动力学，并在此理论分析的基础上提出了一种新的检测算法。最近，我们成功地利用生成泛函分析(GFA)获得了不需要高斯近似的PIC检测动力学的精确解。我们将动态复制理论(DRT)应用于码分多址检测动力学。我们还将这些理论应用到图像、恢复和纠错编码中。我们提出了一种具有足够精度的非平均场模型和有限尺寸模型的数值方法，其中使用主成分分析(PCA)来分析通过蒙特卡罗模拟采样的位形。以两体平均场Sourlas码和三体平均场Sourlas码为例，我们证明了即使对于一般的非MF有限尺寸模型，PCA方法也是有效的。量子退火法利用量子涨落来搜索经典无序哈密顿量的基态，这与最大后验概率估计相对应。如果量子涨落随时间足够缓慢且线性地减小，则量子退火后的剩余能量下降为退火时间的倒数平方。通过对小系统的数值计算，给出了剩余能量的这一特征，并基于量子绝热理论进行了推导，提出了一种新的基于稀疏连接随机网络的有损数据压缩框架，该框架最初用于纠错编码。

We investigated dynamics for CDMA multiuser detection. We applied the statistical neurodynamics, which is based on Gaussian approximation for input distribution, to a detection dynamics of a parallel interference canceller (PIC), and proposed a new detection algorithm based on this theoretical analysis. Recently, we have succeeded in obtaining an exact solution for the PIC detection dynamics without the Gaussian approximation using generating functional analysis (GFA). We applied a dynamical replica theory (DRT) to the CDMA detection dynamics. We also applied these theories to the image, restoration and the error-correcting code.We proposed a numerical approach for a class of non-mean field models and finite-size models with sufficient accuracy, in which principal component analysis (PCA) is employed to analyze configurations sampled through Monte Carlo simulations. We showed that the PCA approach will be effective even with general non-MF finite-size models using examples of two-and three-body mean-field Sourlas codes.The quantum annealing method exploits quantum fluctuations to search the ground state of classical disordered Hamiltonian, which corresponds to maximum a posteriori probability estimation. If the quantum fluctuation is reduced sufficiently slowly and linearly by the time, the residual energy after the quantum annealing falls as the inverse square of the annealing time. We show this feature of the residual energy by numerical calculations for small-sized systems and derive it on the basis of the quantum adiabatic theorem.Additionally, we proposed a new framework for the lossy date compression using a sparsely connected random network, which was originally used on the error-correcting code.