├── README.md ├── SuperLR.pdf ├── index.html ├── ml history.pdf ├── pages ├── SuperLR.css ├── SuperLR.outline ├── SuperLR1.page ├── SuperLR10.page ├── SuperLR11.page ├── SuperLR12.page ├── SuperLR13.page ├── SuperLR14.page ├── SuperLR15.page ├── SuperLR16.page ├── SuperLR17.page ├── SuperLR18.page ├── SuperLR19.page ├── SuperLR2.page ├── SuperLR20.page ├── SuperLR21.page ├── SuperLR22.page ├── SuperLR23.page ├── SuperLR24.page ├── SuperLR25.page ├── SuperLR26.page ├── SuperLR27.page ├── SuperLR28.page ├── SuperLR29.page ├── SuperLR3.page ├── SuperLR30.page ├── SuperLR31.page ├── SuperLR32.page ├── SuperLR33.page ├── SuperLR34.page ├── SuperLR35.page ├── SuperLR36.page ├── SuperLR37.page ├── SuperLR38.page ├── SuperLR39.page ├── SuperLR4.page ├── SuperLR40.page ├── SuperLR41.page ├── SuperLR42.page ├── SuperLR43.page ├── SuperLR44.page ├── SuperLR45.page ├── SuperLR46.page ├── SuperLR47.page ├── SuperLR48.page ├── SuperLR49.page ├── SuperLR5.page ├── SuperLR50.page ├── SuperLR51.page ├── SuperLR52.page ├── SuperLR53.page ├── SuperLR54.page ├── SuperLR55.page ├── SuperLR56.page ├── SuperLR57.page ├── SuperLR58.page ├── SuperLR59.page ├── SuperLR6.page ├── SuperLR60.page ├── SuperLR61.page ├── SuperLR62.page ├── SuperLR63.page ├── SuperLR64.page ├── SuperLR65.page ├── SuperLR66.page ├── SuperLR67.page ├── SuperLR68.page ├── SuperLR69.page ├── SuperLR7.page ├── SuperLR70.page ├── SuperLR71.page ├── SuperLR72.page ├── SuperLR73.page ├── SuperLR74.page ├── SuperLR75.page ├── SuperLR76.page ├── SuperLR77.page ├── SuperLR78.page ├── SuperLR79.page ├── SuperLR8.page ├── SuperLR80.page ├── SuperLR81.page ├── SuperLR82.page ├── SuperLR83.page ├── SuperLR84.page ├── SuperLR85.page ├── SuperLR86.page ├── SuperLR87.page ├── SuperLR88.page ├── SuperLR89.page ├── SuperLR9.page ├── SuperLR90.page ├── SuperLR91.page ├── SuperLR92.page ├── SuperLR93.page ├── SuperLR94.page ├── SuperLR95.page ├── SuperLR96.page ├── base.min.css ├── bg1.png ├── bg11.png ├── bg12.png ├── bg13.png ├── bg15.png ├── bg16.png ├── bg17.png ├── bg18.png ├── bg19.png ├── bg1a.png ├── bg1b.png ├── bg1d.png ├── bg1e.png ├── bg1f.png ├── bg21.png ├── bg22.png ├── bg23.png ├── bg24.png ├── bg25.png ├── bg26.png ├── bg27.png ├── bg28.png ├── bg29.png ├── bg2a.png ├── bg2b.png ├── bg2d.png ├── bg2e.png ├── bg2f.png ├── bg3.png ├── bg30.png ├── bg31.png ├── bg32.png ├── bg33.png ├── bg34.png ├── bg35.png ├── bg36.png ├── bg37.png ├── bg38.png ├── bg39.png ├── bg3b.png ├── bg3d.png ├── bg3e.png ├── bg3f.png ├── bg4.png ├── bg40.png ├── bg41.png ├── bg43.png ├── bg45.png ├── bg46.png ├── bg47.png ├── bg48.png ├── bg49.png ├── bg4a.png ├── bg4b.png ├── bg4c.png ├── bg4d.png ├── bg4e.png ├── bg4f.png ├── bg5.png ├── bg50.png ├── bg51.png ├── bg53.png ├── bg54.png ├── bg55.png ├── bg57.png ├── bg59.png ├── bg5a.png ├── bg5b.png ├── bg5c.png ├── bg5d.png ├── bg5e.png ├── bg5f.png ├── bg6.png ├── bg60.png ├── bg7.png ├── bg8.png ├── bg9.png ├── bgb.png ├── bgc.png ├── bgd.png ├── bge.png ├── bgf.png ├── compatibility.min.js ├── f1.woff ├── f10.woff ├── f11.woff ├── f12.woff ├── f13.woff ├── f14.woff ├── f15.woff ├── f16.woff ├── f17.woff ├── f18.woff ├── f19.woff ├── f1a.woff ├── f1b.woff ├── f1c.woff ├── f1d.woff ├── f1e.woff ├── f1f.woff ├── f2.woff ├── f20.woff ├── f21.woff ├── f22.woff ├── f23.woff ├── f24.woff ├── f25.woff ├── f26.woff ├── f27.woff ├── f28.woff ├── f29.woff ├── f2a.woff ├── f2b.woff ├── f2c.woff ├── f2d.woff ├── f2e.woff ├── f2f.woff ├── f3.woff ├── f30.woff ├── f31.woff ├── f32.woff ├── f33.woff ├── f34.woff ├── f35.woff ├── f36.woff ├── f37.woff ├── f38.woff ├── f39.woff ├── f3a.woff ├── f3b.woff ├── f3c.woff ├── f3d.woff ├── f3e.woff ├── f3f.woff ├── f4.woff ├── f40.woff ├── f41.woff ├── f42.woff ├── f43.woff ├── f44.woff ├── f45.woff ├── f46.woff ├── f47.woff ├── f48.woff ├── f49.woff ├── f4a.woff ├── f4b.woff ├── f4c.woff ├── f4d.woff ├── f4e.woff ├── f4f.woff ├── f5.woff ├── f50.woff ├── f51.woff ├── f52.woff ├── f53.woff ├── f54.woff ├── f55.woff ├── f56.woff ├── f57.woff ├── f58.woff ├── f59.woff ├── f5a.woff ├── f5b.woff ├── f5c.woff ├── f5d.woff ├── f5e.woff ├── f5f.woff ├── f6.woff ├── f60.woff ├── f61.woff ├── f62.woff ├── f63.woff ├── f64.woff ├── f65.woff ├── f66.woff ├── f67.woff ├── f68.woff ├── f69.woff ├── f7.woff ├── f8.woff ├── f9.woff ├── fa.woff ├── fancy.min.css ├── fb.woff ├── fc.woff ├── fd.woff ├── fe.woff ├── ff.woff ├── index.html ├── pdf2htmlEX-64x64.png └── pdf2htmlEX.min.js └── r_chunqi.pdf /README.md: -------------------------------------------------------------------------------- 1 | # SuperLR-Page 2 | 3 | 4 | 5 | 12 | 13 | -------------------------------------------------------------------------------- /SuperLR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/bjsc/SuperLR-Page/8a247e50d349b33696ea69935d6b7341e712132c/SuperLR.pdf -------------------------------------------------------------------------------- /ml history.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/bjsc/SuperLR-Page/8a247e50d349b33696ea69935d6b7341e712132c/ml history.pdf -------------------------------------------------------------------------------- /pages/SuperLR.outline: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /pages/SuperLR1.page: -------------------------------------------------------------------------------- 1 |
Chunqi SHI & Jonah BU
66天写的逻辑回归
The Logistic Regression Written in 66 Days
June 8, 2016
SYZYGY
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脶 脱 腌腯腧腩腳腩腴腩腣回归
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Appendices
脷
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Appendix A
附附附录录录
A.1 Sigmoid函函函数数数, Logistic函函函数数数和和和Logit函函函数数数
A.1.1 Sigmoid函函函数数数
腓腩腧腭腯腩腤函 数 又 称 为S函 数 ， 是 一 种 最 常 用 的 激 发 函 数脨腁腣腴腩腶腡腴腩腯腮 腦腵腮腣脭
腴腩腯腮脩。其他常见的激发函数包括阶跃函数脨腓腴腥腰 腆腵腮腣腴腩腯腮脩，腔腡腮腈函数，修
正线性单元脨腒腥腣腴腩脌腥腤 腌腩腮腥腡腲 腕腮腩腴脬 腒腥腌腕脩。而腓腩腧腭腯腩腤函数应用 之广，又被
称为软阶跃函数脨腓腯腦腴 腓腴腥腰 腆腵腮腣腴腩腯腮脩
S脨t脩 脽
脱
脱 脫 e
−t
.
它一个很大优点是连 续可导，并且导数形式非常好，可以用它本 身的二次
函数来表示。
S
0
脨t脩 脽 S脨t脩脨脱 − S脨t脩脩.
A.1.2 Logistic函函函数数数
逻辑函数脨腌腯腧腩腳腴腩腣 腆腵腮腣腴腩腯腮脩，有称为逻辑曲线值域腛脰, L腝上，k为陡度脨腓腴腥腥腰腮腥腳腳脩，
以x
0
为中值点脨腍腩腤腰腯腩腮腴脩 的S脨腓腩腧腭腯腩腤脩曲线。所以直观上来说，逻辑函数
是腓腩腧腭腯腩腤函数在值域，陡度和中值点上的一个更为一般的形式。
f脨x脩 脽
L
脱 脫 腥
−k(x−x
0
)
而把k 脽 脱, x
0
脽 脰, L 脽 脱的逻辑函数称为标准逻辑函数。所以标准逻辑函
数脨腓腴腡腮腤腡腲腤 腌腯腧腩腳腴腩腣 腆腵腮腣腴腩腯腮脩就是腓腩腧腭腯腩腤函数。
脹
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脱脰 腁 附录
A.1.3 Logit函函函数数数
腌腯腧腩腴函数是对数赔率脨腌腯腧脭腯腤腤腳脩脬 是赔率脨腯腤腤腳脩的自然对数脨腎腡腴腵腲腡腬 腌腯腧腡脭
腲腩腴腨腭脩形式。直观看上去和逻辑函数没有联系。
腬腯腧腩腴脨p脩 脽 腬腯腧
p
脱 − p
脽 腬腯腧脨p脩 − 腬腯腧脨脱 − p脩 脽 −腬腯腧
脱
p
− 脱
.
但是，其实他们关系非常，腌腯腧腩腴函数的反函数就是标准腌腯腧腩腳腴腩腣函数。
腬腯腧腩腳腴腩腣脨α脩 脽 腬腯腧腩腴
−1
脨α脩 脽
脱
脱 脫 腥腸腰脨−α脩
在之后一章，我们还会通过腌腯腧腩腴函数来解释腌腯腧腩腳腴腩腣回归的由来。
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Part II
从从从广广广义义义线线线性性性模模模型型型(GLM)出出出发发发
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脲脮脲 深入理解广义线性模型 脱脷
g
c
脨µ脩 脽 腬腯腧
θ
脱 − θ
该式就是腬腯腧腩腳腩腴腩腣函数。
所以，从广义线性模 型的角度来看，因为期望输出服从二项分布 ，所以我
们选择腬腯腧腩腳腩腴腩腣函数作为标准链接函数，而这就意味着我们选择了腬腯腧腩腳腴腩腣回
归。
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脱脸 脲 广义线性模型
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Appendices
脱脹
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Appendix B
附附附录录录
B.1 指指指数数数分分分布布布族族族补补补充充充
前面我们提到了的概率分布的指数分 布族是指脨Natural Form脬 NF 脩。还有没
有其他形式的描述呢？其实还有更为一般化的形式脨General Form脬 GF 脩，和
均值自然形式脨Mean Value Natural Form脬 MVNF 脩。这里稍微补充一下他们
之间的关系。
B.1.1 一一一般般般形形形式式式(GF)
一般指数分布族脬
p脨y脻 θ脩 脽 腥腸腰腛d脨θ脩e脨y脩 脫 g脨θ脩 脫 h脨y脩腝
在一般形式里面，所有的参数表达式都放在指数函数中，包括三部分
脱脮 自变量y的函数部分h脨y脩
脲脮 参数θ的函数部分g脨θ脩
脳脮 自变量y与参数θ的函数的乘积d脨θ脩e脨y脩
B.1.2 自自自然然然形形形式式式(NF)
指数分布族的自然形式在前面我们提过：
p脨y脻 θ, φ脩 脽 腥腸腰腛
yθ − b脨θ脩
a脨φ脩
脫 c脨y, φ脩腝
脲脱
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Preface
逻辑回归号称实业界 用的最广的机器学习算法之一。本文试图分成两大部分
进行表述：
脱脮 知篇：知为本，知则有亲，有亲可久。
• 第腉部分脺 逻辑回归简介
• 第腉腉部分脺 广义线性模型
• 第腉腉腉部分脺 最小结构风险
• 第腉腖部分脺 腓腯腦腴腭腡腸回归
• 第腖部分脺 最大熵模型
• 第腖腉部分脺 利用熵来解释腇腌腍脬腍腡腸腅腮腴脬腍腌腅
• 第腖腉腉部分脺 逻辑腐腃腁扩展
脲脮 用篇：用为体，用而有功，有功则大。
• 第??部分脺 对比常见算法脨腓腖腍脬腎腂脬腌腄腁脩
• 第??部分脺 优化求解
• 第??部分脺 并行架构
我们计划先发布知篇，在收集反馈修改后再发布用篇。从计划 动笔到 现在我
们一起伏案二个多月，因此这个版本我们命名为《脶脶天写の逻辑回归》。
、
chunqi.shi@hotmail.com; Jonah.Bu@outlook.com
腶
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脲脲 腂 附录
这种形式又称为分散指数分布族脨腄腩腳腰腥腲腳腩腯腮 腅腸腰腯腮腥腮腴腩腡腬 腆腡腭腩腬腹脬 腄腅腆脩：
脱脮 累积生成函数脨腃腵腭腵腬腡腮腴 腇腥腮腥腲腡腴腩腮腧 腆腵腮腣腴腩腯腮脩 b脨θ脩
脲脮 标准参数脨腃腡腮腯腮腩腣腡腬 腐腡腲腡腭腥腴腥腲脩 θ
脳脮 分散参数脨腄腩腳腰腥腲腳腩腯腮 腐腡腲腡腭腥腴腥腲脩 φ
分散指数函数也可以写成λ的形式脬 λ就称为精确参数脨腐腲腥腣腩腳腩腯腮 腐腡腲腡腭腥腴腥腲脩：
令
λ 脽
脱
a脨φ脩
⇒ p脨y脻 θ, λ脩 脽 腥腸腰腛λ{yθ − b脨θ脩} 脫 c脨y, λ脩腝
指数分布族有一些特征前面我们也提到了脺
脱脮 均值脺 µ ≡ E腛Y 腝 脽 b
0
脨θ脩
脲脮 方差脺 V ar腛Y 腝 脽 b
00
脨θ脩φ 脽 V ar脨µ脩φ
脳脮 标准链接函数：g脨x脩函数满足θ ≡ g脨µ脩
B.1.3 均均均值值值自自自然然然形形形式式式(MVNF)
均值自然形式脬 我们把标准参数脨θ脩 换成数学期望µ的函数µ 脽 τ 脨θ脩 那么θ 脽
τ
−1
脨µ脩并且τ
−1
脨x脩 脽 g脨x脩脺
p脨y脻 µ, φ脩 脽 腥腸腰腛
τ
−1
脨µ脩y − b脨τ
−1
脨µ脩脩
a脨φ脩
脫 c脨y, φ脩腝
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Part III
结结结构构构风风风险险险最最最小小小(SRM)的的的视视视角角角
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腶腩 腐腲腥腦腡腣腥
请您给出意见和建议：
• 修改建议：腗腉腋腉
• 问题汇报：腉腓腓腕腅腓
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脳脴 脳 结构风险最小脨腓腒腍脩
Fig. 3.5 腌腡腰腬腡腣腥 腶腳 腎腯腲腭腡腬 腄腩腳腴腲腩腢腵腴腩腯腮
既然L
2
和L
1
分 别 具有不同的优势 ，那么把两者结合起来是不是会更好
呢？这就是腅腬腡腳腴腩腣腎腥腴的想法。从它的图形上来看，它既有比较突出的顶点
在坐标轴上，又接近圆形限制 。这样对共线性和稀疏 化都有很大帮助 。虽
然腅腬腡腳腴腩腣腎腥腴的效果与腌腡腳腳腯非常相似，但是腌腡腳腳腯对共线性是比较敏感的，
会随机选择其中一个特征并把其他特征的系数置脰。而腅腬腡腳腴腩腣腎腥腴则会在两 者
之间选择一个平衡的点进行加权。
Fig. 3.6 腅腬腡腳腴腩腣 腎腥腴
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脴脰 脴 正则化脨腒腥腧腵腬腡腲腩腺腡腴腩腯腮脩
Fig. 4.2 腃腍 腡腮腤 腌腯腳腳 腆腵腮腣腴腩腯腮脨 腚腥腲腯脭腏腮腥 腌腯腳腳脬 腈腩腮腧腥 腌腯腳腳 腡腮腤 腌腯腧腩腳腴腩腣 腌腯腳腳脩
φ脨x脩 脽
脱
腬腯腧 脲
腬腯腧 脱 脫 e
−x
于是我们可以把腌腯腧腩腳腴腩腣 腌腯腳腳和腈腩腮腧腥 腌腯腳腳可以看成是脰脭脱 腌腯腳腳的一个上
限脨参见图脴脮脲脩。我们知道脰脭脱 腌腯腳腳的另外一个常用的上限是指数损失脨Exponential
Loss脩脺
I脨x ≤ 脰脩 ≤ e
−x
⇔ φ脨x脩 脽 e
−x
而这个对数损失对应的这是腁腤腡腢腯腯腳腴算法的损失 函 数脨参见图脴脮脳脩。这个上限
在腁腤腡腢腯腯腳腴误差收敛性证明中也会用到。
Table 4.1 损失函数
损失函数 函数形式 对应算法
脰脭脱损失脨腚腥腲腯脭腏腮腥 腌腯腳腳脩 I脨x ≤ 脰脩 腌腩腮腥腡腲 腂腩腮腡腲腹 腃腬腡腳腳腩脌腣腡腴腩腯腮
铰链损失脨腈腩腮腧腥 腌腯腳腳脩 φ脨x脩 脽 腭腡腸脨脰, 脱 − x脩 腓腖腍
逻辑损失脨腌腯腧腩腳腴腩腣 腌腯腳腳脩 φ脨x脩 脽
1
log 2
腬腯腧 脱 脫 e
−x
腌腯腧腩腳腴腩腣 腒腥腧腲腥腳腳腩腯腮
指数损失脨腅腸腰腯腮腥腮腴腩腡腬 腌腯腳腳脩 φ脨x脩 脽 e
−x
腁腤腡腂腯腯腳腴
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Part IV
从从从两两两类类类(Bi-Class)到到到多多多类类类(Multi-Class)
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脴脴
前面的部分中，我们 分 别从广义线性模型和结构风险最小两个角 度对逻辑
回归进行了推导和解释。逻辑回归是个两类问题的分类算法，如果面对的是
多类问题的分 类算法，应该怎么办 呢？接下来我们要从解决两类问题的逻辑
回归推广到解决多类问题的软最大回归脨Softmax Regression脩
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Chapter 5
从从从二二二项项项(Binomial)分分分布布布到到到多多多
项项项(Multinominal)分分分布布布
5.1 抛抛抛硬硬硬币币币(Coin Toss)和和和掷掷掷骰骰骰子子子(Dice)
5.1.1 伯伯伯努努努力力力(Bernoulli)分分分布布布
对伯努 力脨腂腥腲腮腯腵腬腬腩脩分 布Bernoulli脨p脩进行说明最常见的例子是抛硬币。假
设抛一次硬币正面的概率p，那么反面的概率就为脱 − p，把两 者统一起来，
所以腂腥腲腮腯腵腬腬腩的概率表达式是：
f脨x脻 p脩 脽 p
x
脨脱 − p脩
1−x
x ∈ {脰, 脱}
5.1.2 二二二项项项(Binomial)分分分布布布
如果我们连续的抛脱个硬币n次，那么我们就得到了二项分布B脨n, p脩。所以 二
项分布是一组伯努力脨腂腥腲腮腯腵腬腬腩脩分布变量之和。
X
k
∼ Bernoulli脨p脩 ⇒ Y 脽
n
X
k=1
X
k
∼ 腂脨n, p脩
f脨x脻 n, p脩 脽
n
x
p
x
脨脱 − p脩
n−x
x ∈ {脰, 脱, . . . , n}
可以看出伯努力分布是二项分布的一个特例：
Bernoulli脨p脩 脽 B脨脱, p脩
脴脵
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Part V
最最最大大大熵熵熵(MaxEnt)
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Part VI
从从从熵熵熵(Entropy)解解解释释释GLM, MaxEnt,
MLE
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脶脶 脸 来自熵脨腅腮腴腲腯腰腹脩的解释
I脨X脻 Y 脩 脽
X
x,y
p脨x, y脩 腬腯腧
p脨x, y脩
p脨x脩p脨y脩
脽
X
x,y
p脨x, y脩 腬腯腧
p脨x, y脩
p脨x脩
−
X
x,y
p脨x, y脩 腬腯腧 p脨y脩
脽
X
x,y
p脨x脩p脨y|x脩 腬腯腧 p脨y|x脩 −
X
x,y
p脨x, y脩 腬腯腧 p脨y脩
脽
X
x
p脨x脩
X
y
p脨y|x脩 腬腯腧 p脨y|x脩
!
−
X
y
腬腯腧 p脨y脩
X
x
p脨x, y脩
!
脽 −
X
x
p脨x脩H脨Y |X 脽 x脩 −
X
y
腬腯腧 p脨y脩p脨y脩
脽 −H脨Y |X脩 脫 H脨Y 脩
脽 H脨Y 脩 − H脨Y |X脩.
Fig. 8.2 互信息脨腍腵腴腵腡腬 腉腮腦腯腲腭腡腴腩腯腮脩
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脸脮脲 最大信息熵原理 脷脱
`脨θ脩 脽 腬腯腧 P 脨X|θ脩 脽
N
X
n=1
腬腯腧 p脨x
n
|θ脩
结合起来最终得到：
θ
∗
脽 腡腲腧 腭腡腸
θ
RE 脨腾p脨x脩kp脨x|θ脩脩
脽 腡腲腧 腭腡腸
θ
X
x
腾p脨x脩 腬腯腧 p脨x|θ脩
脽 腡腲腧 腭腡腸
θ
脱
N
N
X
n=1
腬腯腧 p脨x
n
|θ脩
脽 腡腲腧 腭腡腸
θ
脱
N
`脨θ脩
脽 腡腲腧 腭腡腸
θ
`脨θ脩
由此根据最大熵，我们可以得出最大似然估计 。此外根据式子脨脸脮脶脩，我们可
以得到如下公式：
RE 脨腾p脨x脩kp脨x|θ脩脩 脽 −D
KL
脨腾p脨x脩kp脨x|θ脩脩 脽 H脨腾p脨x脩脩 脫
脱
N
`脨θ脩
这个表达式在变分分析脨腖腡腲腩腡腴腩腯腮腡腬 腉腮腦腥腲腥腮腣腥脩可能会来构建逼近下限。
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脷脲 脸 来自熵脨腅腮腴腲腯腰腹脩的解释
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Appendices
脷脳
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脷脶 腃 附录
Fig. C.2 鲁道夫脮克劳修斯腒腵腤腯腬腦脮腃腬腡腵腳腩腵腳
Fig. C.3 约西亚脮吉布斯腊腯腳腩腡腨脮腇腩腢腢腳
Fig. C.4 艾尔文脮薛定谔腅腲腷腩腮脮腓腣腨腲腯腤腩腮腧腥腲
艾尔文脮薛定谔腅腲腷腩腮脮腓腣腨腲腯腤腩腮腧腥腲，同样出生 维也纳大学，量子力 学奠基
人（量子力学量力学）第一次把概率倒数解释称状态数量，这样对数形式就
可以解释成编码长度。并且引入了负号。
年轻的克劳 德脮香农腃腬腡腵腤腥脮腓腨腡腮腮腯腮，密西根大学的天才，将薛定谔的解
释正式引入信号处理中，定义为信息损失。从此建立了信息熵，开创 了信息
理论。在和计算机之神约翰脮冯脮诺伊曼腊腯腨腮脮腶腯腮脮腎腥腵腭腡腮腮，来自 苏黎世联邦
理工大学的少年天才，讨论后正式把信息熵定义成不确信的测度。并且给出
了腈函数的定义。
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腃脮脱 信息熵的由来 脷脷
Fig. C.5 克劳德脮香农腃腬腡腵腤腥脮腓腨腡腮腮腯腮
Fig. C.6 约翰脮冯脮诺伊曼腊腯腨腮脮腶腯腮脮腎腥腵腭腡腮腮
Fig. C.7 埃德温脮杰恩斯腅腤腷腩腮脮腊腡腹腮腥腳
埃德温脮杰恩斯腅腤腷腩腮脮 腔脮 腊腡腹腮腥腳脬 普林斯顿的杰出统计学家，开创 性的 通
过逻辑来解释统计，通过最大熵来解释统计，由 此开创 了统计分析的贝叶斯
学派。
至此，熵一发而不可收拾的成为诸多信息处理理论基石！
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Part VII
Logistic PCA
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Part I
逻逻逻辑辑辑回回回归归归(LR)简简简介介介
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脸脸 脹 腌腯腧腩腳腴腩腣 腐腃腁和腐腃腁在指数分布族上的推广
投影脨Bregman Projection脩，x
i
的腂腲腥腧腭腡腮投影为g脨θ
i
脩；在投影之前 ，我们
把θ
i
表 示 成 一 个 更 低 维 度 空 间 的 基v
1
, . . . , v
l
的 线 性 组 合 ， 其 组 合 系 数
为a
i,1
, . . . , a
i,l
，即θ
i
脽
P
l
k=1
a
ik
v
k
；最终的目标是找到A和V，使得x
i
与
其腂腲腥腧腭腡腮投影的腂腲腥腧腭腡腮散度最小，即：
脨V, A脩 脽 腡腲腧 腭腩腮
V,A
n
X
i=1
B
F
脨x
i
kg脨
l
X
k=1
a
i,k
v
k
脩脩 脨脹脮脴脶脩
从V的角度看，指数分 布族上的腐腃腁是在一个低维空间里寻找一组基脨矩
阵V脩，该组基定义了一个尽可能接近 原始数据x
i
的面脨腓腵腲腦腡腣腥脩。设由 基V定
义的面为S脨V脩 脽 {g脨aV脩|a ∈ R
l
}，则我们寻找的V为：
V 脽 腡腲腧 腭腩腮
V
X
i
腭腩腮
q∈S(V)
B
F
脨x
i
kq脩 脨脹脮脴脷脩
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