From 11d182272a589b8f14d435ee2d90d6268720a6c8 Mon Sep 17 00:00:00 2001 From: benjas <909336740@qq.com> Date: Tue, 4 May 2021 12:07:07 +0800 Subject: [PATCH] Add.Support vector machines --- .../7.支持向量机——升维打击.md | 62 +++++++++++------- .../assets/1620100449792.png | Bin 0 -> 76149 bytes .../assets/1620100488418.png | Bin 0 -> 9951 bytes .../assets/1620100528671.png | Bin 0 -> 5525 bytes .../assets/1620100544902.png | Bin 0 -> 9298 bytes .../assets/1620100565725.png | Bin 0 -> 14077 bytes .../assets/1620100585894.png | Bin 0 -> 9565 bytes .../assets/1620100641814.png | Bin 0 -> 120688 bytes .../assets/1620100681735.png | Bin 0 -> 22331 bytes .../assets/1620100936402.png | Bin 0 -> 78136 bytes .../assets/1620101031289.png | Bin 0 -> 78635 bytes .../assets/1620101185820.png | Bin 0 -> 142109 bytes 12 files changed, 39 insertions(+), 23 deletions(-) create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100449792.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100488418.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100528671.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100544902.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100565725.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100585894.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100641814.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100681735.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620100936402.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620101031289.png create mode 100644 机器学习算法原理及推导/李航——统计学习方法/assets/1620101185820.png diff --git a/机器学习算法原理及推导/李航——统计学习方法/7.支持向量机——升维打击.md b/机器学习算法原理及推导/李航——统计学习方法/7.支持向量机——升维打击.md index c956715..339d346 100644 --- a/机器学习算法原理及推导/李航——统计学习方法/7.支持向量机——升维打击.md +++ b/机器学习算法原理及推导/李航——统计学习方法/7.支持向量机——升维打击.md @@ -139,13 +139,8 @@ Support vector machines > 图中心,虚线到实线的距离我们称之为γ,我们要做的是最大化γ,使得这个超平面调整为γ的一个最大值,等价于找到了最优的超平面 **式子如下:** -$$ -\max_{w,b} \quad γ -$$ -$$ -s.t.\quad y_i(\frac{w}{||w||}*x_i+\frac{b}{||w||})≥γ \quad i=1,2,...,N -$$ +![1620100488418](assets/1620100488418.png) > γ:表示几何间隔 > @@ -164,47 +159,68 @@ $$ 既然我们最终是\frac{\hat{γ}}{||w||} ,那么式子我们可以简化成 $$ -$$ -y_i(wx_i+b)≥\hat{γ},其中\hat{γ}是函数间隔 -$$ +![1620100528671](assets/1620100528671.png) max的时候是几何间隔,也就是最终s.t. 还是会约束着它朝着几何间隔去走,但是这样的好处就是下方的||w||就没有了 **简化后如下:** -$$ -\max_{w,b} \quad \frac{\hat{γ}}{||w||} -$$ -$$ -s.t.\quad y_i(w*x_i+b)≥γ \quad i=1,2,...,N -$$ +![1620100544902](assets/1620100544902.png) 之前我们说过,对于函数间隔,我们等比例放大缩小w、b可以让最终结果变成1,也就是γ=1 **再简化后:** + +![1620100565725](assets/1620100565725.png) $$ -\max_{w,b} \quad \frac{1}{||w||} +后面要用到拉格朗日乘子法,我们把\frac{1}{||w||}变成\frac{1}{2}||w||^2,这两者是等价的 $$ +**再简化后:** + +![1620100585894](assets/1620100585894.png) $$ -s.t.\quad y_i(w*x_i+b)≥1 \quad i=1,2,...,N +\min_{w,b} \quad \frac{1}{2}||w||^2 $$ - $$ -我们想要最大化\frac{1}{||w||},那么相当于最小化||w|| +s.t.\quad y_i(w*x_i+b)-1≥0 \quad i=1,2,...,N $$ +> 利用拉格朗日乘子法,推导成如下式子 + +![1620100681735](assets/1620100681735.png) $$ -后面要用到拉格朗日乘子法,我们把\frac{1}{||w||}变成\frac{1}{2}||w||^2,这两者是等价的 +L(w,b,α)=\quad \frac{1}{2}||w||^2-\sum^N_{i=1}α_iy_i(w*x_i+b)+\sum^N_{i=1}α_i $$ -**再简化后:** $$ -\min_{w,b} \quad \frac{1}{2}||w||^2 +目标:\min_{w,b}\max_aL(w,b,α) $$ $$ -s.t.\quad y_i(w*x_i+b)-1≥0 \quad i=1,2,...,N +转换成:\max_a\min_{w,b}L(w,b,α) $$ + + +将拉格朗日函数L(w,b,α)分别对w,b求偏导并令其等于0 + +![1620100449792](assets/1620100449792.png) + +进行推导 + +![1620100641814](assets/1620100641814.png) + + + +2.求minL(w,b,α)对α的极大,即是对偶问题 + +![1620100936402](assets/1620100936402.png) + +3.求max转换成min: + +![1620101031289](assets/1620101031289.png) + +接下来就是求解α的问题了 + diff --git a/机器学习算法原理及推导/李航——统计学习方法/assets/1620100449792.png b/机器学习算法原理及推导/李航——统计学习方法/assets/1620100449792.png new file mode 100644 index 0000000000000000000000000000000000000000..cf0c514d8b6a1cccefade234a7de2c1e4acdff06 GIT binary patch literal 76149 zcmV)Px#1ZP1_K>z@;j|==^1poj532;bRa{vGmbN~PnbOGLGA9w%&|D{PpK~#8N?EUw5 zWl56fi5c*qH6nnBKmWmQ&XWu=OY%+Oj31hjqt`2Cn&f4F!7ghc3P4~Fa4ue-Uq zxw*NyU%&TO{rCUPV=XEs+gXH`{IrjX6eaWv!qtgaF87==BFARnIv9r9sdZjKo8 zMatP(vqC{(9laP`o~d4#=W!s)=YN#D7(7T>`P{+gjE<4htE8YrHcw39kYT|JrN{)i zL?wXBD#YM`oJi$I$x4!vjOVP71P*u^;-gvuLw=rC?WGcd;BP*CzO-^k*F?#IbLdrP z<=rc9jw6BRjP&|P_F2g)aWfB`r5I8#yjl(^MLL$Jo~MAMB9!U^ELMVZIaly&0M}Jz z=h=bF3VXb;l)n_1htM%vPi>gBna$0Lp5mNF%2Z`D>MUF@&dgd(b-`+CYSa(1md^_3 zf8q6*EJej;VbUw5Ti>yye|2g z3S6+9=i+ifhm_4x{}4(jQCzL6Rtm|Q;!?WhgeX6Z+EN&)md1QqeuyblJB#=FDBgvX z)bJ=n};lyW0G~k zg(0hc%W)o}Trlwi!heci9HR>_g^82IdpV@*agDQiOl6dZX+zh^3_YwXPK0w( zDZ-mjNZ8Nkla#Oo-hL6sfLK>oXZ3XrR;W=sdtMAoLGiWL)YxcEO--IAvL?;~5^dzj zwGy~o1H3F<^UL`(IY{~E@Qyet4uhL-U%*L3a7oCg;fNIZpW@>zAjPHl1aOf5MT&XA z=tg&fdhi)EqV-aUWBfemeQsV>Pa2mO3U9srgP%|lC+!-?QS}n05pV%ye1sSR(h=Y? 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