本書全面敘述了蒙特卡羅方法，包括序貫蒙特卡羅方法、馬爾可夫鏈蒙特卡羅方法基礎(chǔ)、Metropolis算法及其變體、吉布斯采樣器及其變體、聚類采樣方法、馬爾可夫鏈蒙特卡羅的收斂性分析、數(shù)據(jù)驅(qū)動(dòng)的馬爾可夫鏈蒙特卡羅方法、哈密頓和朗之萬蒙特卡羅方法、隨機(jī)梯度學(xué)習(xí)和可視化能級(jí)圖等。為了便于學(xué)習(xí)，每章都包含了不同領(lǐng)域的代表性應(yīng)用實(shí)例。本書旨在統(tǒng)計(jì)學(xué)和計(jì)算機(jī)科學(xué)之間架起一座橋梁以彌合它們之間的鴻溝，以便將其應(yīng)用于計(jì)算機(jī)視覺、計(jì)算機(jī)圖形學(xué)、機(jī)器學(xué)習(xí)、機(jī)器人學(xué)、人工智能等領(lǐng)域解決更廣泛的問題，同時(shí)使這些領(lǐng)域的科學(xué)家和工程師們更容易地利用蒙特卡羅方法加強(qiáng)他們的研究。
朱松純，1996年獲得哈佛大學(xué)計(jì)算機(jī)科學(xué)博士學(xué)位，現(xiàn)任北京通用人工智能研究院院長、北京大學(xué)人工智能研究院院長、北京大學(xué)講席教授、清華大學(xué)基礎(chǔ)科學(xué)講席教授；曾任美國加州大學(xué)洛杉磯分校（UCLA）統(tǒng)計(jì)學(xué)與計(jì)算機(jī)科學(xué)教授，加州大學(xué)洛杉磯分校視覺、認(rèn)知、學(xué)習(xí)與自主機(jī)器人中心主任。 他長期致力于為視覺和智能探尋一個(gè)統(tǒng)一的統(tǒng)計(jì)與計(jì)算框架：包括作為學(xué)習(xí)與推理的統(tǒng)一表達(dá)和數(shù)字蒙特卡洛方法的時(shí)空因果與或圖（STC-AOG）。他在計(jì)算機(jī)視覺、統(tǒng)計(jì)學(xué)習(xí)、認(rèn)知、人工智能和自主機(jī)器人領(lǐng)域發(fā)表了400多篇學(xué)術(shù)論文。他曾獲得了多項(xiàng)榮譽(yù)，2003年因圖像解析的工作成就獲馬爾獎(jiǎng)，1999年因紋理建模、2007年因物體建模兩次獲得馬爾獎(jiǎng)提名。2001 年，他獲得了NSF青年科學(xué)家獎(jiǎng)、ONR青年研究員獎(jiǎng)和斯隆獎(jiǎng)。因?yàn)樵谝曈X模式的概念化、建模、學(xué)習(xí)和推理的統(tǒng)一基礎(chǔ)方面的貢獻(xiàn)，他2008年獲得了國際模式識(shí)別協(xié)會(huì)授予的J.K. Aggarwal獎(jiǎng)。2013 年，他關(guān)于圖像分割的論文獲得了亥姆霍茲獎(jiǎng)（Helmholtz Test-of-Time Award）。2017年，他因生命度建模工作獲國際認(rèn)知學(xué)會(huì)計(jì)算建模獎(jiǎng)。2011年，他當(dāng)選IEEE Fellow。他兩次擔(dān)任國際計(jì)算機(jī)視覺與模式識(shí)別大會(huì)（CVPR 2012，2019）主席。作為項(xiàng)目負(fù)責(zé)人，他領(lǐng)導(dǎo)了多個(gè)ONR MURI和DARPA團(tuán)隊(duì)，從事統(tǒng)一數(shù)學(xué)框架下的場(chǎng)景和事件理解以及認(rèn)知機(jī)器人的工作。巴布·艾俊，2000 年獲得俄亥俄州立大學(xué)數(shù)學(xué)博士學(xué)位，2005 年獲得加州大學(xué)洛杉磯分校計(jì)算機(jī)科學(xué)博士學(xué)位（師從朱松純博士）。2005年至2007年，他在西門子研究院從事醫(yī)學(xué)成像研究工作，從開始擔(dān)任研究科學(xué)家到后來升任項(xiàng)目經(jīng)理。由于在邊緣空間學(xué)習(xí)方面的工作成就，他與西門子的合作者獲得了2011年Thomas A. Edison專利獎(jiǎng)。2007年，他加入佛羅里達(dá)州立大學(xué)統(tǒng)計(jì)系，從助理教授到副教授，再到2019年擔(dān)任教授。他發(fā)表了70多篇關(guān)于計(jì)算機(jī)視覺、機(jī)器學(xué)習(xí)和醫(yī)學(xué)成像方面的論文，并擁有超過25項(xiàng)與醫(yī)學(xué)成像和圖像去噪相關(guān)的專利。
魏平，西安交通大學(xué)人工智能學(xué)院教授、博士生導(dǎo)師，人工智能學(xué)院副院長，國家級(jí)青年人才，陜西高校青年創(chuàng)新團(tuán)隊(duì)（自主智能系統(tǒng)）帶頭人，西安交通大學(xué)“青年拔尖人才支持計(jì)劃”A類入選者。西安交通大學(xué)學(xué)士、博士學(xué)位，美國加州大學(xué)洛杉磯分校（UCLA）博士后、聯(lián)合培養(yǎng)博士。研究領(lǐng)域包括計(jì)算機(jī)視覺、機(jī)器學(xué)習(xí)、智能系統(tǒng)等。主持國家自然科學(xué)基金項(xiàng)目、國家重點(diǎn)研發(fā)計(jì)劃子課題等科研項(xiàng)目十余項(xiàng)，作為骨干成員參與國家自然科學(xué)基金重大科學(xué)研究計(jì)劃等課題多項(xiàng)。在TPAMI、CVPR、ICCV、ACM MM、AAAI、IJCAI等國際權(quán)威期刊和會(huì)議發(fā)表學(xué)術(shù)論文多篇，是十余個(gè)國際著名期刊和會(huì)議審稿人。擔(dān)任中國自動(dòng)化學(xué)會(huì)網(wǎng)聯(lián)智能專委會(huì)副主任委員、中國圖象圖形學(xué)學(xué)會(huì)機(jī)器視覺專委會(huì)委員。
目 錄
第1 章 蒙特卡羅方法簡(jiǎn)介··············································································.1
1.1 引言·······························································································.1
1.2 動(dòng)機(jī)和目標(biāo)······················································································.1
1.3 蒙特卡羅計(jì)算中的任務(wù)·······································································.2
1.3.1 任務(wù)1：采樣和模擬········································································.3
1.3.2 任務(wù)2：通過蒙特卡羅模擬估算未知量···················································.5
1.3.3 任務(wù)3：優(yōu)化和貝葉斯推理································································.7
1.3.4 任務(wù)4：學(xué)習(xí)和模型估計(jì)···································································.8
1.3.5 任務(wù)5：可視化能級(jí)圖·····································································.9
本章參考文獻(xiàn)··························································································13
第2 章 序貫蒙特卡羅方法··············································································14
2.1 引言·······························································································14
2.2 一維密度采樣···················································································14
2.3 重要性采樣和加權(quán)樣本·······································································15
2.4 序貫重要性采樣（SIS） ······································································18
2.4.1 應(yīng)用：表達(dá)聚合物生長的自避游走························································18
2.4.2 應(yīng)用：目標(biāo)跟蹤的非線性/粒子濾波·······················································20
2.4.3 SMC 方法框架總結(jié)·········································································23
2.5 應(yīng)用：利用SMC 方法進(jìn)行光線追蹤·······················································24
2.6 在重要性采樣中保持樣本多樣性···························································25
2.6.1 基本方法····················································································25
2.6.2 Parzen 窗討論··············································································28
2.7 蒙特卡羅樹搜索················································································29
2.7.1 純蒙特卡羅樹搜索··········································································30
2.7.2 AlphaGo ·····················································································32
2.8 本章練習(xí)·························································································33
本章參考文獻(xiàn)··························································································35
第3 章 馬爾可夫鏈蒙特卡羅方法基礎(chǔ)·······························································36
3.1 引言·······························································································36
蒙特卡羅方法與人工智能
·X ·
3.2 馬爾可夫鏈基礎(chǔ)················································································37
3.3 轉(zhuǎn)移矩陣的拓?fù)洌哼B通與周期······························································38
3.4 Perron-Frobenius 定理··········································································41
3.5 收斂性度量······················································································42
3.6 連續(xù)或異構(gòu)狀態(tài)空間中的馬爾可夫鏈·····················································44
3.7 各態(tài)遍歷性定理················································································45
3.8 通過模擬退火進(jìn)行MCMC 優(yōu)化·····························································46
3.9 本章練習(xí)·························································································49
本章參考文獻(xiàn)··························································································51
第4 章 Metropolis 算法及其變體······································································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 其他接受概率設(shè)計(jì)··········································································55
4.2.4 Metropolis 算法設(shè)計(jì)中的關(guān)鍵問題·························································55
4.3 獨(dú)立Metropolis 采樣···········································································55
4.3.1 IMS 的特征結(jié)構(gòu)············································································56
4.3.2 有限空間的一般首中時(shí)·····································································57
4.3.3 IMS 擊中時(shí)分析············································································57
4.4 可逆跳躍和跨維MCMC ······································································59
4.4.1 可逆跳躍····················································································59
4.4.2 簡(jiǎn)單例子：一維圖像分割··································································60
4.5 應(yīng)用：計(jì)算人數(shù)················································································63
4.5.1 標(biāo)值點(diǎn)過程模型············································································64
4.5.2 MCMC 推理·················································································64
4.5.3 結(jié)果·························································································65
4.6 應(yīng)用：家具布置················································································65
4.7 應(yīng)用：場(chǎng)景合成················································································67
4.8 本章練習(xí)·························································································71
本章參考文獻(xiàn)··························································································72
第5 章 吉布斯采樣器及其變體········································································73
5.1 引言·······························································································73
5.2 吉布斯采樣器···················································································74
目 錄
·XI·
5.2.1 吉布斯采樣器介紹··········································································74
5.2.2 吉布斯采樣器的一個(gè)主要問題·····························································75
5.3 吉布斯采樣器擴(kuò)展·············································································76
5.3.1 擊中逃跑····················································································77
5.3.2 廣義吉布斯采樣器··········································································77
5.3.3 廣義擊中逃跑···············································································77
5.3.4 利用輔助變量采樣··········································································78
5.3.5 模擬退火····················································································78
5.3.6 切片采樣····················································································79
5.3.7 數(shù)據(jù)增強(qiáng)····················································································80
5.3.8 Metropolized 吉布斯采樣器·································································80
5.4 數(shù)據(jù)關(guān)聯(lián)和數(shù)據(jù)增強(qiáng)··········································································82
5.5 Julesz 系綜和MCMC 紋理采樣······························································83
5.5.1 Julesz 系綜：紋理的數(shù)學(xué)定義······························································84
5.5.2 吉布斯系綜和系綜等價(jià)性··································································85
5.5.3 Julesz 系綜采樣·············································································86
5.5.4 實(shí)驗(yàn)：對(duì)Julesz 系綜進(jìn)行采樣·····························································87
5.6 本章練習(xí)·························································································89
本章參考文獻(xiàn)··························································································90
第6 章 聚類采樣方法····················································································91
6.1 引言·······························································································91
6.2 Potts 模型和SW 算法·········································································92
6.3 SW 算法詳解····················································································94
6.3.1 解釋1：Metropolis-Hastings 觀點(diǎn)··························································94
6.3.2 解釋2：數(shù)據(jù)增強(qiáng)··········································································97
6.4 SW 算法的相關(guān)理論結(jié)果··································································.100
6.5 任意概率的SW 切分算法·································································.102
6.5.1 步驟一：數(shù)據(jù)驅(qū)動(dòng)的聚類·······························································.102
6.5.2 步驟二：顏色翻轉(zhuǎn)·······································································.103
6.5.3 步驟三：接受翻轉(zhuǎn)·······································································.104
6.5.4 復(fù)雜性分析···············································································.105
6.6 聚類采樣方法的變體·······································································.106
6.6.1 聚類吉布斯采樣：“擊中逃跑”觀點(diǎn)·····················································.106
6.6.2 多重翻轉(zhuǎn)方案············································································.107
6.7 應(yīng)用：圖像分割·············································································.107
蒙特卡羅方法與人工智能
·X II·
6.8 多重網(wǎng)格和多級(jí)SW 切分算法···························································.110
6.8.1 多重網(wǎng)格SW 切分算法··································································.111
6.8.2 多級(jí)SW 切分算法·······································································.113
6.9 子空間聚類···················································································.114
6.9.1 通過SW 切分算法進(jìn)行子空間聚類·····················································.115
6.9.2 應(yīng)用：稀疏運(yùn)動(dòng)分割····································································.117
6.10 C 4：聚類合作競(jìng)爭(zhēng)約束··································································.121
6.10.1 C 4 算法綜述············································································.123
6.10.2 圖形、耦合和聚類······································································.124
6.10.3 平面圖上的C 4 算法····································································.128
6.10.4 在平面圖上的實(shí)驗(yàn)······································································.131
6.10.5 棋盤Ising 模型·········································································.132
6.10.6 分層圖上的C 4··········································································.136
6.10.7 C 4 分層實(shí)驗(yàn)············································································.138
6.11 本章練習(xí)·····················································································.139
本章參考文獻(xiàn)·······················································································.140
第7 章 MCMC 的收斂性分析·······································································.144
7.1 引言····························································································.144
7.2 關(guān)鍵收斂問題················································································.144
7.3 實(shí)用的監(jiān)測(cè)方法·············································································.145
7.4 洗牌的耦合方法·············································································.146
7.4.1 置頂洗牌·················································································.147
7.4.2 Riffle 洗牌················································································.147
7.5 幾何界限、瓶頸和連通率·································································.149
7.5.1 幾何收斂·················································································.149
7.5.2 交易圖（轉(zhuǎn)換圖）·······································································.150
7.5.3 瓶頸······················································································.150
7.5.4 連通率····················································································.151
7.6 Peskun 有序和遍歷性定理·································································.152
7.7 路徑耦合和精確采樣·······································································.153
7.7.1 從過去耦合···············································································.154
7.7.2 應(yīng)用：對(duì)Ising 模型進(jìn)行采樣···························································.155
7.8 本章練習(xí)······················································································.157
本章參考文獻(xiàn)·······················································································.159
目 錄
·XIII·
第8 章 數(shù)據(jù)驅(qū)動(dòng)的馬爾可夫鏈蒙特卡羅方法···················································.160
8.1 引言····························································································.160
8.2 圖像分割和DDMCMC 方法概述························································.160
8.3 DDMCMC 方法解釋········································································.161
8.3.1 MCMC 方法設(shè)計(jì)的基本問題····························································.163
8.3.2 計(jì)算原子空間中的提議概率：原子粒子················································.164
8.3.3 計(jì)算對(duì)象空間中的提議概率：對(duì)象粒子················································.166
8.3.4 計(jì)算多個(gè)不同的解：場(chǎng)景粒子··························································.167
8.3.5 Ψ-世界實(shí)驗(yàn)··············································································.167
8.4 問題表達(dá)和圖像建模·······································································.168
8.4.1 用于分割的貝葉斯公式··································································.169
8.4.2 先驗(yàn)概率·················································································.169
8.4.3 灰度圖像的似然·········································································.169
8.4.4 模型校準(zhǔn)·················································································.171
8.4.5 彩色圖像模型············································································.172
8.5 解空間分析···················································································.173
8.6 使用遍歷馬爾可夫鏈探索解空間························································.174
8.6.1 五類馬爾可夫鏈動(dòng)態(tài)過程·······························································.174
8.6.2 瓶頸問題·················································································.175
8.7 數(shù)據(jù)驅(qū)動(dòng)方法················································································.176
8.7.1 方法一：原子空間中的聚類·····························································.176
8.7.2 方法二：邊緣檢測(cè)·······································································.180
8.8 計(jì)算重要性提議概率·······································································.180
8.9 計(jì)算多個(gè)不同的解··········································································.183
8.9.1 動(dòng)機(jī)和數(shù)學(xué)原理·········································································.183
8.9.2 用于多種解的K-冒險(xiǎn)家算法····························································.184
8.10 圖像分割實(shí)驗(yàn)···············································································.185
8.11 應(yīng)用：圖像解析············································································.188
8.11.1 自上而下和自下而上的處理···························································.190
8.11.2 生成和判別方法········································································.190
8.11.3 馬爾可夫鏈核和子核···································································.191
8.11.4 DDMCMC 和提議概率·································································.193
8.11.5 馬爾可夫鏈子核········································································.200
8.11.6 圖像解析實(shí)驗(yàn)···········································································.207
8.12 本章練習(xí)·····················································································.210
蒙特卡羅方法與人工智能
·X IV·
本章參考文獻(xiàn)·······················································································.211
第9 章 哈密頓和朗之萬蒙特卡羅方法····························································.215
9.1 引言····························································································.215
9.2 哈密頓力學(xué)···················································································.215
9.2.1 哈密頓方程···············································································.215
9.2.2 HMC 的簡(jiǎn)單模型········································································.216
9.3 哈密頓力學(xué)的性質(zhì)··········································································.217
9.3.1 能量守恒·················································································.217
9.3.2 可逆性····················································································.218
9.3.3 辛結(jié)構(gòu)和體積保持·······································································.219
9.4 哈密頓方程的蛙跳離散化·································································.220
9.4.1 歐拉方法·················································································.220
9.4.2 改良的歐拉方法·········································································.220
9.4.3 蛙跳積分器···············································································.221
9.4.4 蛙跳積分器的特性·······································································.222
9.5 哈密頓蒙特卡羅方法和朗之萬蒙特卡羅方法·········································.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 調(diào)參················································································.228
9.5.5 HMC 的細(xì)致平衡證明···································································.229
9.6 黎曼流形HMC···············································································.230
9.6.1 HMC 中的線性變換·····································································.230
9.6.2 RMHMC 動(dòng)力學(xué)·········································································.233
9.6.3 RMHMC 算法和變體····································································.235
9.6.4 RMHMC 中的協(xié)方差函數(shù)·······························································.236
9.7 HMC 實(shí)踐·····················································································.237
9.7.1 受約束正態(tài)分布的模擬實(shí)驗(yàn)·····························································.237
9.7.2 使用RMHMC 對(duì)邏輯回歸系數(shù)進(jìn)行采樣···············································.241
9.7.3 使用LMC 采樣圖像密度：FRAME、GRADE 和DeepFRAME ·······················.243
9.8 本章練習(xí)······················································································.248
本章參考文獻(xiàn)·······················································································.249
第10 章 隨機(jī)梯度學(xué)習(xí)················································································.250
10.1 引言···························································································.250
目 錄
·XV·
10.2 隨機(jī)梯度：動(dòng)機(jī)和性質(zhì)···································································.250
10.2.1 引例·····················································································.251
10.2.2 Robbins-Monro 定理····································································.253
10.2.3 隨機(jī)梯度下降和朗之萬方程···························································.254
10.3 馬爾可夫隨機(jī)場(chǎng)（MRF）模型的參數(shù)估計(jì)···········································.257
10.3.1 利用隨機(jī)梯度學(xué)習(xí)FRAME 模型······················································.258
10.3.2 FRAME 的替代學(xué)習(xí)方法·······························································.259
10.3.3 FRAME 算法的四種變體·······························································.261
10.3.4 紋理分析實(shí)驗(yàn)···········································································.264
10.4 用神經(jīng)網(wǎng)絡(luò)學(xué)習(xí)圖像模型································································.267
10.4.1 對(duì)比發(fā)散與持續(xù)對(duì)比發(fā)散······························································.267
10.4.2 使用深度網(wǎng)絡(luò)學(xué)習(xí)圖像的勢(shì)能模型：DeepFRAME···································.268
10.4.3 生成器網(wǎng)絡(luò)和交替反向傳播···························································.271
10.4.4 協(xié)作網(wǎng)絡(luò)和生成器模型································································.275
10.5 本章練習(xí)·····················································································.279
本章參考文獻(xiàn)·······················································································.279
第11 章 可視化能級(jí)圖················································································.282
11.1 引言···························································································.282
11.2 能級(jí)圖的示例、結(jié)構(gòu)和任務(wù)·····························································.282
11.2.1 基于能量的狀態(tài)空間劃分······························································.285
11.2.2 構(gòu)造非連通圖（DG）··································································.286
11.2.3 二維ELM 示例·········································································.287
11.2.4 表征學(xué)習(xí)任務(wù)的難度（或復(fù)雜度）····················································.289
11.3 廣義Wang-Landau 算法··································································.290
11.3.1 GWL 映射的能壘估計(jì)··································································.291
11.3.2 用GWL 估算體積······································································.292
11.3.3 GWL 收斂性分析·······································································.294
11.4 GWL 實(shí)驗(yàn)···················································································.295
11.4.1 高斯混合模型的GWL 映射····························································.295
11.4.2 語法模型的GWL 映射·································································.301
11.5 用吸引-擴(kuò)散可視化能級(jí)圖······························································.305
11.5.1 亞穩(wěn)定性和宏觀劃分···································································.306
11.5.2 吸引-擴(kuò)散簡(jiǎn)介·········································································.307
11.5.3 吸引-擴(kuò)散和Ising 模型································································.309
11.5.4 吸引-擴(kuò)散ELM 算法（ADELM 算法）···············································.311
蒙特卡羅方法與人工智能
·X VI·
11.5.5 調(diào)優(yōu)ADELM ···········································································.313
11.5.6 AD 能壘估計(jì)···········································································.314
11.6 用GWL 和ADELM 可視化SK 自旋玻璃模型······································.315
11.7 使用吸引?擴(kuò)散可視化圖像空間························································.318
11.7.1 圖像星系的結(jié)構(gòu)········································································.318
11.7.2 可視化實(shí)驗(yàn)·············································································.319
11.8 本章練習(xí)·····················································································.324
本章參考文獻(xiàn)·······················································································.324
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