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蒙特卡罗方法与人工智能 读者对象:本书适合计算机、人工智能、机器人等领域的教师、学生阅读和参考,也适合相关领域的研究者和工业界的从业者阅读。
本书全面叙述了蒙特卡罗方法,包括序贯蒙特卡罗方法、马尔可夫链蒙特卡罗方法基础、Metropolis算法及其变体、吉布斯采样器及其变体、聚类采样方法、马尔可夫链蒙特卡罗的收敛性分析、数据驱动的马尔可夫链蒙特卡罗方法、哈密顿和朗之万蒙特卡罗方法、随机梯度学习和可视化能级图等。为了便于学习,每章都包含了不同领域的代表性应用实例。本书旨在统计学和计算机科学之间架起一座桥梁以弥合它们之间的鸿沟,以便将其应用于计算机视觉、计算机图形学、机器学习、机器人学、人工智能等领域解决更广泛的问题,同时使这些领域的科学家和工程师们更容易地利用蒙特卡罗方法加强他们的研究。
朱松纯,1996年获得哈佛大学计算机科学博士学位,现任北京通用人工智能研究院院长、北京大学人工智能研究院院长、北京大学讲席教授、清华大学基础科学讲席教授;曾任美国加州大学洛杉矶分校(UCLA)统计学与计算机科学教授,加州大学洛杉矶分校视觉、认知、学习与自主机器人中心主任。 他长期致力于为视觉和智能探寻一个统一的统计与计算框架:包括作为学习与推理的统一表达和数字蒙特卡洛方法的时空因果与或图(STC-AOG)。他在计算机视觉、统计学习、认知、人工智能和自主机器人领域发表了400多篇学术论文。他曾获得了多项荣誉,2003年因图像解析的工作成就获马尔奖,1999年因纹理建模、2007年因物体建模两次获得马尔奖提名。2001 年,他获得了NSF青年科学家奖、ONR青年研究员奖和斯隆奖。因为在视觉模式的概念化、建模、学习和推理的统一基础方面的贡献,他2008年获得了国际模式识别协会授予的J.K. Aggarwal奖。2013 年,他关于图像分割的论文获得了亥姆霍兹奖(Helmholtz Test-of-Time Award)。2017年,他因生命度建模工作获国际认知学会计算建模奖。2011年,他当选IEEE Fellow。他两次担任国际计算机视觉与模式识别大会(CVPR 2012,2019)主席。作为项目负责人,他领导了多个ONR MURI和DARPA团队,从事统一数学框架下的场景和事件理解以及认知机器人的工作。巴布·艾俊,2000 年获得俄亥俄州立大学数学博士学位,2005 年获得加州大学洛杉矶分校计算机科学博士学位(师从朱松纯博士)。2005年至2007年,他在西门子研究院从事医学成像研究工作,从开始担任研究科学家到后来升任项目经理。由于在边缘空间学习方面的工作成就,他与西门子的合作者获得了2011年Thomas A. Edison专利奖。2007年,他加入佛罗里达州立大学统计系,从助理教授到副教授,再到2019年担任教授。他发表了70多篇关于计算机视觉、机器学习和医学成像方面的论文,并拥有超过25项与医学成像和图像去噪相关的专利。
魏平,西安交通大学人工智能学院教授、博士生导师,人工智能学院副院长,国家级青年人才,陕西高校青年创新团队(自主智能系统)带头人,西安交通大学“青年拔尖人才支持计划”A类入选者。西安交通大学学士、博士学位,美国加州大学洛杉矶分校(UCLA)博士后、联合培养博士。研究领域包括计算机视觉、机器学习、智能系统等。主持国家自然科学基金项目、国家重点研发计划子课题等科研项目十余项,作为骨干成员参与国家自然科学基金重大科学研究计划等课题多项。在TPAMI、CVPR、ICCV、ACM MM、AAAI、IJCAI等国际权威期刊和会议发表学术论文多篇,是十余个国际著名期刊和会议审稿人。担任中国自动化学会网联智能专委会副主任委员、中国图象图形学学会机器视觉专委会委员。
目 录
第1 章 蒙特卡罗方法简介··············································································.1 1.1 引言·······························································································.1 1.2 动机和目标······················································································.1 1.3 蒙特卡罗计算中的任务·······································································.2 1.3.1 任务1:采样和模拟········································································.3 1.3.2 任务2:通过蒙特卡罗模拟估算未知量···················································.5 1.3.3 任务3:优化和贝叶斯推理································································.7 1.3.4 任务4:学习和模型估计···································································.8 1.3.5 任务5:可视化能级图·····································································.9 本章参考文献··························································································13 第2 章 序贯蒙特卡罗方法··············································································14 2.1 引言·······························································································14 2.2 一维密度采样···················································································14 2.3 重要性采样和加权样本·······································································15 2.4 序贯重要性采样(SIS) ······································································18 2.4.1 应用:表达聚合物生长的自避游走························································18 2.4.2 应用:目标跟踪的非线性/粒子滤波·······················································20 2.4.3 SMC 方法框架总结·········································································23 2.5 应用:利用SMC 方法进行光线追踪·······················································24 2.6 在重要性采样中保持样本多样性···························································25 2.6.1 基本方法····················································································25 2.6.2 Parzen 窗讨论··············································································28 2.7 蒙特卡罗树搜索················································································29 2.7.1 纯蒙特卡罗树搜索··········································································30 2.7.2 AlphaGo ·····················································································32 2.8 本章练习·························································································33 本章参考文献··························································································35 第3 章 马尔可夫链蒙特卡罗方法基础·······························································36 3.1 引言·······························································································36 蒙特卡罗方法与人工智能 ·X · 3.2 马尔可夫链基础················································································37 3.3 转移矩阵的拓扑:连通与周期······························································38 3.4 Perron-Frobenius 定理··········································································41 3.5 收敛性度量······················································································42 3.6 连续或异构状态空间中的马尔可夫链·····················································44 3.7 各态遍历性定理················································································45 3.8 通过模拟退火进行MCMC 优化·····························································46 3.9 本章练习·························································································49 本章参考文献··························································································51 第4 章 Metropolis 算法及其变体······································································52 4.1 引言·······························································································52 4.2 Metropolis-Hastings 算法······································································52 4.2.1 原始Metropolis-Hastings 算法······························································53 4.2.2 Metropolis-Hastings 算法的另一形式·······················································54 4.2.3 其他接受概率设计··········································································55 4.2.4 Metropolis 算法设计中的关键问题·························································55 4.3 独立Metropolis 采样···········································································55 4.3.1 IMS 的特征结构············································································56 4.3.2 有限空间的一般首中时·····································································57 4.3.3 IMS 击中时分析············································································57 4.4 可逆跳跃和跨维MCMC ······································································59 4.4.1 可逆跳跃····················································································59 4.4.2 简单例子:一维图像分割··································································60 4.5 应用:计算人数················································································63 4.5.1 标值点过程模型············································································64 4.5.2 MCMC 推理·················································································64 4.5.3 结果·························································································65 4.6 应用:家具布置················································································65 4.7 应用:场景合成················································································67 4.8 本章练习·························································································71 本章参考文献··························································································72 第5 章 吉布斯采样器及其变体········································································73 5.1 引言·······························································································73 5.2 吉布斯采样器···················································································74 目 录 ·XI· 5.2.1 吉布斯采样器介绍··········································································74 5.2.2 吉布斯采样器的一个主要问题·····························································75 5.3 吉布斯采样器扩展·············································································76 5.3.1 击中逃跑····················································································77 5.3.2 广义吉布斯采样器··········································································77 5.3.3 广义击中逃跑···············································································77 5.3.4 利用辅助变量采样··········································································78 5.3.5 模拟退火····················································································78 5.3.6 切片采样····················································································79 5.3.7 数据增强····················································································80 5.3.8 Metropolized 吉布斯采样器·································································80 5.4 数据关联和数据增强··········································································82 5.5 Julesz 系综和MCMC 纹理采样······························································83 5.5.1 Julesz 系综:纹理的数学定义······························································84 5.5.2 吉布斯系综和系综等价性··································································85 5.5.3 Julesz 系综采样·············································································86 5.5.4 实验:对Julesz 系综进行采样·····························································87 5.6 本章练习·························································································89 本章参考文献··························································································90 第6 章 聚类采样方法····················································································91 6.1 引言·······························································································91 6.2 Potts 模型和SW 算法·········································································92 6.3 SW 算法详解····················································································94 6.3.1 解释1:Metropolis-Hastings 观点··························································94 6.3.2 解释2:数据增强··········································································97 6.4 SW 算法的相关理论结果··································································.100 6.5 任意概率的SW 切分算法·································································.102 6.5.1 步骤一:数据驱动的聚类·······························································.102 6.5.2 步骤二:颜色翻转·······································································.103 6.5.3 步骤三:接受翻转·······································································.104 6.5.4 复杂性分析···············································································.105 6.6 聚类采样方法的变体·······································································.106 6.6.1 聚类吉布斯采样:“击中逃跑”观点·····················································.106 6.6.2 多重翻转方案············································································.107 6.7 应用:图像分割·············································································.107 蒙特卡罗方法与人工智能 ·X II· 6.8 多重网格和多级SW 切分算法···························································.110 6.8.1 多重网格SW 切分算法··································································.111 6.8.2 多级SW 切分算法·······································································.113 6.9 子空间聚类···················································································.114 6.9.1 通过SW 切分算法进行子空间聚类·····················································.115 6.9.2 应用:稀疏运动分割····································································.117 6.10 C 4:聚类合作竞争约束··································································.121 6.10.1 C 4 算法综述············································································.123 6.10.2 图形、耦合和聚类······································································.124 6.10.3 平面图上的C 4 算法····································································.128 6.10.4 在平面图上的实验······································································.131 6.10.5 棋盘Ising 模型·········································································.132 6.10.6 分层图上的C 4··········································································.136 6.10.7 C 4 分层实验············································································.138 6.11 本章练习·····················································································.139 本章参考文献·······················································································.140 第7 章 MCMC 的收敛性分析·······································································.144 7.1 引言····························································································.144 7.2 关键收敛问题················································································.144 7.3 实用的监测方法·············································································.145 7.4 洗牌的耦合方法·············································································.146 7.4.1 置顶洗牌·················································································.147 7.4.2 Riffle 洗牌················································································.147 7.5 几何界限、瓶颈和连通率·································································.149 7.5.1 几何收敛·················································································.149 7.5.2 交易图(转换图)·······································································.150 7.5.3 瓶颈······················································································.150 7.5.4 连通率····················································································.151 7.6 Peskun 有序和遍历性定理·································································.152 7.7 路径耦合和精确采样·······································································.153 7.7.1 从过去耦合···············································································.154 7.7.2 应用:对Ising 模型进行采样···························································.155 7.8 本章练习······················································································.157 本章参考文献·······················································································.159 目 录 ·XIII· 第8 章 数据驱动的马尔可夫链蒙特卡罗方法···················································.160 8.1 引言····························································································.160 8.2 图像分割和DDMCMC 方法概述························································.160 8.3 DDMCMC 方法解释········································································.161 8.3.1 MCMC 方法设计的基本问题····························································.163 8.3.2 计算原子空间中的提议概率:原子粒子················································.164 8.3.3 计算对象空间中的提议概率:对象粒子················································.166 8.3.4 计算多个不同的解:场景粒子··························································.167 8.3.5 Ψ-世界实验··············································································.167 8.4 问题表达和图像建模·······································································.168 8.4.1 用于分割的贝叶斯公式··································································.169 8.4.2 先验概率·················································································.169 8.4.3 灰度图像的似然·········································································.169 8.4.4 模型校准·················································································.171 8.4.5 彩色图像模型············································································.172 8.5 解空间分析···················································································.173 8.6 使用遍历马尔可夫链探索解空间························································.174 8.6.1 五类马尔可夫链动态过程·······························································.174 8.6.2 瓶颈问题·················································································.175 8.7 数据驱动方法················································································.176 8.7.1 方法一:原子空间中的聚类·····························································.176 8.7.2 方法二:边缘检测·······································································.180 8.8 计算重要性提议概率·······································································.180 8.9 计算多个不同的解··········································································.183 8.9.1 动机和数学原理·········································································.183 8.9.2 用于多种解的K-冒险家算法····························································.184 8.10 图像分割实验···············································································.185 8.11 应用:图像解析············································································.188 8.11.1 自上而下和自下而上的处理···························································.190 8.11.2 生成和判别方法········································································.190 8.11.3 马尔可夫链核和子核···································································.191 8.11.4 DDMCMC 和提议概率·································································.193 8.11.5 马尔可夫链子核········································································.200 8.11.6 图像解析实验···········································································.207 8.12 本章练习·····················································································.210 蒙特卡罗方法与人工智能 ·X IV· 本章参考文献·······················································································.211 第9 章 哈密顿和朗之万蒙特卡罗方法····························································.215 9.1 引言····························································································.215 9.2 哈密顿力学···················································································.215 9.2.1 哈密顿方程···············································································.215 9.2.2 HMC 的简单模型········································································.216 9.3 哈密顿力学的性质··········································································.217 9.3.1 能量守恒·················································································.217 9.3.2 可逆性····················································································.218 9.3.3 辛结构和体积保持·······································································.219 9.4 哈密顿方程的蛙跳离散化·································································.220 9.4.1 欧拉方法·················································································.220 9.4.2 改良的欧拉方法·········································································.220 9.4.3 蛙跳积分器···············································································.221 9.4.4 蛙跳积分器的特性·······································································.222 9.5 哈密顿蒙特卡罗方法和朗之万蒙特卡罗方法·········································.223 9.5.1 HMC 建模················································································.223 9.5.2 HMC 算法················································································.224 9.5.3 LMC 算法················································································.226 9.5.4 HMC 调参················································································.228 9.5.5 HMC 的细致平衡证明···································································.229 9.6 黎曼流形HMC···············································································.230 9.6.1 HMC 中的线性变换·····································································.230 9.6.2 RMHMC 动力学·········································································.233 9.6.3 RMHMC 算法和变体····································································.235 9.6.4 RMHMC 中的协方差函数·······························································.236 9.7 HMC 实践·····················································································.237 9.7.1 受约束正态分布的模拟实验·····························································.237 9.7.2 使用RMHMC 对逻辑回归系数进行采样···············································.241 9.7.3 使用LMC 采样图像密度:FRAME、GRADE 和DeepFRAME ·······················.243 9.8 本章练习······················································································.248 本章参考文献·······················································································.249 第10 章 随机梯度学习················································································.250 10.1 引言···························································································.250 目 录 ·XV· 10.2 随机梯度:动机和性质···································································.250 10.2.1 引例·····················································································.251 10.2.2 Robbins-Monro 定理····································································.253 10.2.3 随机梯度下降和朗之万方程···························································.254 10.3 马尔可夫随机场(MRF)模型的参数估计···········································.257 10.3.1 利用随机梯度学习FRAME 模型······················································.258 10.3.2 FRAME 的替代学习方法·······························································.259 10.3.3 FRAME 算法的四种变体·······························································.261 10.3.4 纹理分析实验···········································································.264 10.4 用神经网络学习图像模型································································.267 10.4.1 对比发散与持续对比发散······························································.267 10.4.2 使用深度网络学习图像的势能模型:DeepFRAME···································.268 10.4.3 生成器网络和交替反向传播···························································.271 10.4.4 协作网络和生成器模型································································.275 10.5 本章练习·····················································································.279 本章参考文献·······················································································.279 第11 章 可视化能级图················································································.282 11.1 引言···························································································.282 11.2 能级图的示例、结构和任务·····························································.282 11.2.1 基于能量的状态空间划分······························································.285 11.2.2 构造非连通图(DG)··································································.286 11.2.3 二维ELM 示例·········································································.287 11.2.4 表征学习任务的难度(或复杂度)····················································.289 11.3 广义Wang-Landau 算法··································································.290 11.3.1 GWL 映射的能垒估计··································································.291 11.3.2 用GWL 估算体积······································································.292 11.3.3 GWL 收敛性分析·······································································.294 11.4 GWL 实验···················································································.295 11.4.1 高斯混合模型的GWL 映射····························································.295 11.4.2 语法模型的GWL 映射·································································.301 11.5 用吸引-扩散可视化能级图······························································.305 11.5.1 亚稳定性和宏观划分···································································.306 11.5.2 吸引-扩散简介·········································································.307 11.5.3 吸引-扩散和Ising 模型································································.309 11.5.4 吸引-扩散ELM 算法(ADELM 算法)···············································.311 蒙特卡罗方法与人工智能 ·X VI· 11.5.5 调优ADELM ···········································································.313 11.5.6 AD 能垒估计···········································································.314 11.6 用GWL 和ADELM 可视化SK 自旋玻璃模型······································.315 11.7 使用吸引?扩散可视化图像空间························································.318 11.7.1 图像星系的结构········································································.318 11.7.2 可视化实验·············································································.319 11.8 本章练习·····················································································.324 本章参考文献·······················································································.324
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