Markdown & LaTex 常用语法

目录

• blog 的目录
• 博客园自带目录
• 用 javascript 自定义目录
• 自定义目录：页面定制CSS代码
• 自定义目录：页脚Html代码
• 主标题
  • 副标题
• h1，一级标题
• h2，二级标题
• h3，三级标题
• h4，四级标题
  • h5，五级标题
    • h6，六级标题
• 注释
• 常用的符号及文本形式
• 如果你想在markdown中文本缩进
• 无序列表
• 有序列表
• 连接跳转
• 插入图像
• 斜体，加粗，分割线
• 单行和多行代码块
• 使用 ag{n} 为公式添加编号
• markdown 的表格 和 LaTeX 中的空格
• 例子汇总

blog 的目录

博客园自带目录

[TOC]
出现在随笔页面的开始处，可以帮你显示目录，而无需自己配置javascript，但对比下，和自己配置的略有不同，自定义的有 Back to the top，Go to page bottom，目录编号

用 javascript 自定义目录

自定义目录：页面定制CSS代码

<style type="text/css">
#cnblogs_post_body
{
    color: black;
    font: 0.875em/1.5em "微软雅黑" , "PTSans" , "Arial" ,sans-serif;
    font-size: 16px;
}
#cnblogs_post_body h2    {
    background: #2B6695;
    border-radius: 6px 6px 6px 6px;
    box-shadow: 0 0 0 1px #5F5A4B, 1px 1px 6px 1px rgba(10, 10, 0, 0.5);
    color: #FFFFFF;
    font-family: "微软雅黑" , "宋体" , "黑体" ,Arial;
    font-size: 17px;
    font-weight: bold;
    height: 25px;
    line-height: 25px;
    margin: 18px 0 !important;
    padding: 8px 0 5px 5px;
    text-shadow: 2px 2px 3px #222222;
}
#cnblogs_post_body h3{
    background: #2B6600;
    border-radius: 6px 6px 6px 6px;
    box-shadow: 0 0 0 1px #5F5A4B, 1px 1px 6px 1px rgba(10, 10, 0, 0.5);
    color: #FFFFFF;
    font-family: "微软雅黑" , "宋体" , "黑体" ,Arial;
    font-size: 13px;
    font-weight: bold;
    height: 24px;
    line-height: 23px;
    margin: 12px 0 !important;
    padding: 5px 0 5px 20px;
    text-shadow: 2px 2px 3px #222222;
}
#cnblogs_post_body a {
    color: #21759b;
    transition-delay: 0s;
    transition-duration: 0.4s;
    transition-property: all;
    transition-timing-function: linear;
}
#cnblogs_post_body a:hover{
    margin-left: 10px
}

#navCategory a{
    display: block;
    transition: all 1s;
    
}
#navCategory a:hover{
    margin-left: 10px
}

#blog-sidecolumn  a{
    display: block;
    transition:all 1s;
}
#blog-sidecolumn a:hover{
    margin-left: 10px
}

#sidebar_toptags li a{
    float:left;
}
#TopViewPostsBlock li a{
    margin-left: 5px;
}
#cnblogs_post_body a{
    display: inline-block;
    transition:all 1s;
}
</style>

自定义目录：页脚Html代码

<script language="javascript" type="text/javascript">
// Generate a directory index list
// ref: http://www.cnblogs.com/wangqiguo/p/4355032.html
// ref: https://www.cnblogs.com/xuehaoyue/p/6650533.html
// modified by: keyshaw
function GenerateContentList()
{
    var mainContent = $('#cnblogs_post_body');

    //If your chapter title isn't `h2`, You just replace the h2 here.
    var h2_list = $('#cnblogs_post_body h2');
    // var go_to_bottom = '<div style="text-align: right;"><a href="#_page_bottom" style="color:#f68a33">Go to page bottom</a></div>';
    var bottom_label = '<div style="text-align: right;"><a href="#_labelTop" style="color:#f68a33">Back to the top</a><a name="_page_bottom"></a></div>'

    if(mainContent.length < 1)
        return;

    if(h2_list.length>0)
    {
        var content = '<div style="text-align: right;"><a href="#_page_bottom" style="color:#f68a33">Go to page bottom</a></div><a name="_labelTop"></a>';
        content += '<div id="navCategory" style="color:#152e97;">';
        // coutent += '<div style="text-align: right;"><a href="#_page_bottom" style="color:#f68a33">Go to page bottom</a></div>'
        content += '<h1 style="font-size:16px;background: #f68a33;border-radius: 6px 6px 6px 6px;box-shadow: 0 0 0 1px #5F5A4B, 1px 1px 6px 1px rgba(10, 10, 0, 0.5);color: #FFFFFF;font-size: 17px;font-weight: bold;height: 25px;line-height: 25px;margin: 18px 0 !important;padding: 8px 0 5px 30px;"><b>Catalogue</b></h1>';
        // ol - ordered; ul - unordered
        content += '<ol>';
        for(var i=0; i<h2_list.length; i++)
        {
            // add 'Back to the top' before h2
            var go_to_top_2 = '<div style="text-align: right;"><a href="#_labelTop" style="color:#f68a33">Back to the top</a><a name="_label' + i + '"></a></div>';
            $(h2_list[i]).before(go_to_top_2);
            
            var h3_list = $(h2_list[i]).nextAll("h3");
            
            var li3_content = '';
            for(var j=0; j<h3_list.length; j++)
            {

                var tmp_3 = $(h3_list[j]).prevAll('h2').first();
                if(!tmp_3.is(h2_list[i]))
                    break;

                var go_to_top_3 = '<div style="text-align: right;"><a href="#_labelTop" style="color:#f68a33">Back to the top</a><a name="_label' + i + '_' + j + '"></a></div>';
                $(h3_list[j]).before(go_to_top_3);

                // li3_content += '<li><a href="#_label' + i + '_' + j + '"style="font-size:12px;color:#2b6695;">' + $(h3_list[j]).text() + '</a></li>';

                var li4_content = '';
                var h4_list = $(h3_list[j]).nextAll("h4");
                for(var k=0; k<h4_list.length; k++)
                {
                    var tmp_4 = $(h4_list[k]).prevAll('h3').first();
                    if(!tmp_4.is(h3_list[j]))
                        break;

                    var go_to_top_4 = '<div style="text-align: right;"><a href="#_labelTop" style="color:#f68a33">Back to the top</a><a name="_label' + i + '_' + j + '_' + k + '"></a></div>';
                    $(h4_list[k]).before(go_to_top_4);

                    li4_content += '<li><a href="#_label' + i + '_' + j + '_' + k + '"style="font-size:12px;color:#2b6695;">' + $(h4_list[k]).text() + '</a></li>';
                }

                
                if(li4_content.length > 0)
                    li3_content += '<li><a href="#_label' + i + '_' + j + '"style="font-size:12px;color:#2b6695;">' + $(h3_list[j]).text() + '</a><ul>' + li4_content + '</ul></li>';
                else
                    li3_content += '<li><a href="#_label' + i + '_' + j + '"style="font-size:12px;color:#2b6695;">' + $(h3_list[j]).text() + '</a></li>';

            }

            var li2_content = '';
            if(li3_content.length > 0)
                li2_content = '<li><a href="#_label' + i + '"style="font-size:12px;color:#2b6695;">' + $(h2_list[i]).text() + '</a><ul>' + li3_content + '</ul></li>';
            else
                li2_content = '<li><a href="#_label' + i + '"style="font-size:12px;color:#2b6695;">' + $(h2_list[i]).text() + '</a></li>';
            content += li2_content;

        }
        content += '</ol>';
        content += '</div><p>&nbsp;</p>';
        content += '<hr />';

        // $(mainContent[0]).prepend(go_to_bottom);
        $(mainContent[0]).prepend(content);
        $(mainContent[0]).append(bottom_label);
    }
}

GenerateContentList();
</script>

主标题
===

主标题

副标题
---

副标题

# h1，一级标题

h1，一级标题

# h2，二级标题

h2，二级标题

# h3，三级标题

h3，三级标题

# h4，四级标题

h4，四级标题

## h5，五级标题

h5，五级标题

### h6，六级标题

h6，六级标题

注释

><space><space><enter>
这是一段注释
<space><space><enter>
**a** : 这是一段注释
<space><space><enter>
**b** : 这是一段注释

这是一段注释

a : 这是一段注释

b : 这是一段注释

>这是一段注释
**a** : 这是一段注释
**b** : 这是一段注释

这是一段注释
a : 这是一段注释
b : 这是一段注释

常用的符号及文本形式

$underset{sim}{A}$ : (underset{sim}{Λ})
$widehat{y}$ : (widehat{y})
<u>我被下划线了</u> : 我被下划线了
~~我被删除线了~~ : 我被删除线了
$mathrm{d}a$ : (mathrm{d}a)
$da$ : (da)
$( ig( Big( igg( Bigg($ : (( ig( Big( igg( Bigg()

useful links for your LaTeX:
https://en.wikibooks.org/wiki/LaTeX/Mathematics
https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference
https://zh.numberempire.com/latexequationeditor.php

如果你想在markdown中文本缩进

   here
     here
         here

   here
     here
         here

无序列表

<ul>
    <li>无序列表1</li>
    <li>无序列表2</li>
    <li>无序列表3</li>
</ul>

• 无序列表1
• 无序列表2
• 无序列表3

* 无序列表1
* 无序列表2
* 无序列表3

• 无序列表1
• 无序列表2
• 无序列表3

+ 无序列表4
+ 无序列表5
+ 无序列表6

• 无序列表4
• 无序列表5
• 无序列表6

- 无序列表7
- 无序列表8
- 无序列表9

• 无序列表7
• 无序列表8
• 无序列表9

* 呆萌小二郎
  * 23岁
  * 前端工程师
  喜欢看书，撸代码，写博客...
* 呆萌小二郎2
  * 嘻嘻哈哈
    * 开心
* 呆萌小二郎3

• 呆萌小二郎
  • 23岁
  • 前端工程师
    喜欢看书，撸代码，写博客...
• 呆萌小二郎2
  • 嘻嘻哈哈
    • 开心
• 呆萌小二郎3

有序列表

<ol>
    <li>有序列表1</li>
    <li>有序列表2</li>
    <li>有序列表3</li>
</ol>

1. 有序列表1
2. 有序列表2
3. 有序列表3

1. 有序列表1
2. 有序列表2
3. 有序列表3

1. 有序列表1
2. 有序列表2
3. 有序列表3

1. 有序列表1
1. 有序列表2
1. 有序列表3

1. 有序列表1
2. 有序列表2
3. 有序列表3

连接跳转

[呆萌小二郎博客跳转链接](http://blog.zhouminghang.xyz)
呆萌小二郎博客跳转链接

度娘一下，你就知道： <http://www.baidu.com>
度娘一下，你就知道： http://www.baidu.com

<http://blog.zhouminghang.xyz>
http://blog.zhouminghang.xyz

插入图像

![xxx](https://timgsa.baidu.com/timg?image&quality=80&size=b9999_10000&sec=1553421507058&di=5171700a9aefd5831ce01b6c0f341436&imgtype=0&src=http%3A%2F%2Fpic175.nipic.com%2Ffile%2F20180711%2F24144945_161350611036_2.jpg)

斜体，加粗，分割线

*斜体写法1* 和 _斜体写法2_
斜体写法1 和 斜体写法2

**加粗写法1** 和 __加粗写法2__
加粗写法1 和 加粗写法2

* * *

---

***

---

*****************

---

- - -

---

-----------------

---

---

---

单行和多行代码块

`单行代码`
单行代码

```
多行代码(
这里用来转义符号，
类似于html中单双引号多层嵌套要转义
)
```

多行代码(
        这里用来转义符号，
        类似于html中单双引号多层嵌套要转义
        )

使用 ag{n} 为公式添加编号

不使用 egin{align} 和 end{align} 也可以为公式添加标号，可以使用 ag{n}

$aaa 	ag{1}$
$bbb 	ag{2}$

(aaa ag{1})
(bbb ag{2})

markdown 的表格 和 LaTeX 中的空格

However, this doesn't give the correct result.
LaTeX doesn't respect the white-space left in the code to signify that the y and the dx are independent entities.
Instead, it lumps them altogether.
A quad would clearly be overkill in this situation—what is needed are some small spaces to be utilized in this type of instance, and that's what LaTeX provides:

Command|Description|        Size
:---|---|---:
\,|small space|3/18 of a quad
\:|medium space|4/18 of a quad
\;|large space|5/18 of a quad
\!|negative space|-3/18 of a quad

Command	Description	Size
\,	small space	3/18 of a quad
:	medium space	4/18 of a quad
;	large space	5/18 of a quad
!	negative space	-3/18 of a quad

**Expected Output**:
<table style="100%">
<tr>
<td> **sigmoid_derivative([1,2,3])**</td>
<td> [ 0.19661193  0.10499359  0.04517666] </td>
</tr>
</table>

Expected Output:

**sigmoid_derivative([1,2,3])**

[ 0.19661193 0.10499359 0.04517666]

例子汇总

$$
egin{align} 
& underset{w,b}{mathrm{max}} ;; underset{i}{mathrm{min}} ;; frac{2}{||w||} | w^{	op} x_i + b |, \ 
& mathrm{s.t.} ;;y_i(w^{	op}x_i + b) > 0, ; i = 1,2,...,m.  
onumber 
end{align}
$$

[egin{align} & underset{w,b}{mathrm{max}} ;; underset{i}{mathrm{min}} ;; frac{2}{||w||} | w^{ op} x_i + b |, \ & mathrm{s.t.} ;;y_i(w^{ op}x_i + b) > 0, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
h(x_i)=
egin{cases}
 1 & 	ext{若  }y_i=1; \
 -1 & 	ext{若  }y_i=-1. \
end{cases}
end{align}
$$

(egin{align}h(x_i)=egin{cases} 1 & ext{若 }y_i=1; \ -1 & ext{若 }y_i=-1. \end{cases}end{align})

---

$$
egin{align}
h(x_i) := mathrm{sign}(w^T x_i + b), 	ext{ 其中 } w_i in mathbb{R}^d,b in mathbb{R}.
end{align}
$$

(egin{align}h(x_i) := mathrm{sign}(w^T x_i + b), ext{ 其中 } w_i in mathbb{R}^d,b in mathbb{R}.end{align})

---

$$
egin{align} 
forall_{i.} ;; y_i(w^T x_i + b) > 0 
end{align}
$$

(egin{align} forall_{i.} ;; y_i(w^T x_i + b) > 0 end{align})

---

$$
egin{align}
 y_i h(x_i) = 1 Leftrightarrow y_i mathrm{sign}(w^T x_i + b) = 1  Leftrightarrow y_i(w^T x_i + b) > 0 
end{align}
$$

(egin{align} y_i h(x_i) = 1 Leftrightarrow y_i mathrm{sign}(w^T x_i + b) = 1 Leftrightarrow y_i(w^T x_i + b) > 0 end{align})

---

$$
egin{align}
 frac{1}{||w||} | w^{	op} p + b |
end{align}
$$

(egin{align} frac{1}{||w||} | w^{ op} p + b |end{align})

---

$$
egin{align}
 w^{	op}(x_1-x_2) = w^{	op}x_1-w^{	op}x_2=(-b)-(-b)=0,
end{align}
$$

(egin{align} w^{ op}(x_1-x_2) = w^{ op}x_1-w^{ op}x_2=(-b)-(-b)=0,end{align})

---

(w perp (x_1 - x_2))

---

$$
egin{align}
 mathrm{proj}_w(p-x) 
 &= ||p-x|| cdot |cos (w, p - x)| 
onumber\ 
 &= ||p-x|| cdot frac{|w^{	op}(p-x)|}{||w|| cdot ||p-x||} 
onumber \
 &= frac{1}{||w||} |w^{	op}p - w^{	op}x| 
onumber \ 
 &= frac{1}{||w||} | w^{	op}p + b | 
end{align}
$$

(egin{align} mathrm{proj}_w(p-x) &= ||p-x|| cdot |cos (w, p - x)| onumber\ &= ||p-x|| cdot frac{|w^{ op}(p-x)|}{||w|| cdot ||p-x||} onumber \ &= frac{1}{||w||} |w^{ op}p - w^{ op}x| onumber \ &= frac{1}{||w||} | w^{ op}p + b | end{align})

---

$$
gamma := 2 ; underset{i}{mathrm{min}} frac{1}{||w||} | w^{	op} x_i + b |
$$

(gamma := 2 ; underset{i}{mathrm{min}} frac{1}{||w||} | w^{ op} x_i + b |)

---

$$
egin{align} 
& underset{u}{mathrm{min}} ;; frac{1}{2} u^{	op}Qu+t^{	op}u \ 
& mathrm{s.t.} ;; c_i^{	op} u geq d_i, ; i = 1,2,...,m.  
onumber 
end{align}
$$

[egin{align} & underset{u}{mathrm{min}} ;; frac{1}{2} u^{ op}Qu+t^{ op}u \ & mathrm{s.t.} ;; c_i^{ op} u geq d_i, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align} 
& underset{u}{mathrm{min}} ;; frac{1}{2} u^{	op}Qu+t^{	op}u \ 
& mathrm{s.t.} ;; c_i^{	op} u geq d_i, ; i = 1,2,...,m.  
onumber 
end{align}
$$

[egin{align} & underset{u}{mathrm{min}} ;; frac{1}{2} u^{ op}Qu+t^{ op}u \ & mathrm{s.t.} ;; c_i^{ op} u geq d_i, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
frac{2}{||rw^*||} | (rw^*)^{	op} x_i + rb^* | = frac{2}{||w^*||} | w^{*	op} x_i + b^* |, \
y_i ig( (rw^*)^{	op} x_i + rb^* ig) > 0 Leftrightarrow  y_i (w^{*	op} x_i + b^*)>0.
end{align}
$$

[egin{align} frac{2}{||rw^*||} | (rw^*)^{ op} x_i + rb^* | = frac{2}{||w^*||} | w^{* op} x_i + b^* |, \ y_i ig( (rw^*)^{ op} x_i + rb^* ig) > 0 Leftrightarrow y_i (w^{* op} x_i + b^*)>0. end{align} ]

---

$$
egin{align}
underset{i}{mathrm{min}} ; | w^{	op} x_i + b | = 1.
end{align}
$$

[egin{align} underset{i}{mathrm{min}} ; | w^{ op} x_i + b | = 1. end{align} ]

---

$$
egin{align} 
& underset{w, b}{mathrm{min}} ;; frac{1}{2} w^{	op} w \ 
& mathrm{s.t.} ;; y_i (w^{	op} x_i +b ) geq 1, ; i = 1,2,...,m.  
onumber 
end{align}
$$

[egin{align} & underset{w, b}{mathrm{min}} ;; frac{1}{2} w^{ op} w \ & mathrm{s.t.} ;; y_i (w^{ op} x_i +b ) geq 1, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align} 
& underset{w, b}{mathrm{min}} ;; frac{1}{2} w^{	op} w \ 
& mathrm{s.t.} ;; underset{i}{mathrm{min}} ;;  y_i (w^{	op} x_i +b ) = 1. 
onumber 
end{align}
$$

[egin{align} & underset{w, b}{mathrm{min}} ;; frac{1}{2} w^{ op} w \ & mathrm{s.t.} ;; underset{i}{mathrm{min}} ;; y_i (w^{ op} x_i +b ) = 1. onumber end{align} ]

---

$$
egin{align} 
underset{w, b}{mathrm{argmin}} ;; frac{1}{2} w^{	op} w 
& = underset{w, b}{mathrm{argmin}} ;; frac{1}{2} ||w|| 
onumber\
& = underset{w, b}{mathrm{argmax}} ;; frac{2}{||w||} cdot 1 
onumber\
& = underset{w, b}{mathrm{argmax}} ;; Big( underset{i}{mathrm{min}} ; frac{2}{||w||} y_i (w^{	op} x_i + b) Big) 
onumber\
& = underset{w, b}{mathrm{argmax}} ;; Big( underset{i}{mathrm{min}} ; frac{2}{||w||} |w^{	op} x_i + b| Big) 
end{align}
$$

[egin{align} underset{w, b}{mathrm{argmin}} ;; frac{1}{2} w^{ op} w & = underset{w, b}{mathrm{argmin}} ;; frac{1}{2} ||w|| onumber\ & = underset{w, b}{mathrm{argmax}} ;; frac{2}{||w||} cdot 1 onumber\ & = underset{w, b}{mathrm{argmax}} ;; Big( underset{i}{mathrm{min}} ; frac{2}{||w||} y_i (w^{ op} x_i + b) Big) onumber\ & = underset{w, b}{mathrm{argmax}} ;; Big( underset{i}{mathrm{min}} ; frac{2}{||w||} |w^{ op} x_i + b| Big) end{align} ]

---

$$
egin{align}
u := egin{bmatrix}  w \ b end{bmatrix}, Q := egin{bmatrix}  I&0\0&0 end{bmatrix}, t := 0, \
c_i := y_i egin{bmatrix}  x_i \ 1 end{bmatrix}, d_i := 1,
end{align}
$$

[egin{align} u := egin{bmatrix} w \ b end{bmatrix}, Q := egin{bmatrix} I&0\0&0 end{bmatrix}, t := 0, \ c_i := y_i egin{bmatrix} x_i \ 1 end{bmatrix}, d_i := 1, end{align} ]

---

$$
egin{align}
underset{u}{mathrm{min}} &;; f(u) &\
mathrm{s.t.} &;; g_i (u) leq 0, &i = 1,2,...,m, 
onumber\
                     & ;; h_j (u) = 0, &j = 1,2,...,n, 
onumber
end{align}
$$

[egin{align} underset{u}{mathrm{min}} &;; f(u) &\ mathrm{s.t.} &;; g_i (u) leq 0, &i = 1,2,...,m, onumber\ & ;; h_j (u) = 0, &j = 1,2,...,n, onumber end{align} ]

---

$$
egin{align}
mathcal{L}(u,alpha,eta) := f(u) + sumlimits_{i=1}^{m} alpha_i g_i (u) + sumlimits_{j=1}^{n} eta_j h_j (u)
end{align}
$$

[egin{align} mathcal{L}(u,alpha,eta) := f(u) + sumlimits_{i=1}^{m} alpha_i g_i (u) + sumlimits_{j=1}^{n} eta_j h_j (u) end{align} ]

---

$$
egin{align}
underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;;& mathcal{L} (u, alpha, eta) \
mathrm{s.t.} ;;;;;;& alpha_i geq 0, ;; i = 1,2,...,m. 
onumber
end{align}
$$

[egin{align} underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;;& mathcal{L} (u, alpha, eta) \ mathrm{s.t.} ;;;;;;& alpha_i geq 0, ;; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
& underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;; mathcal{L} (u, alpha, eta) 
onumber \
= & underset{u}{mathrm{min}} Bigg( f(u) + underset{alpha, eta}{mathrm{max}} Big(  sumlimits_{i=1}^{m} alpha_i g_i (u) + sumlimits_{j=1}^{n} eta_j h_j (u) Big)Bigg) 
onumber \
= & underset{u}{mathrm{min}} Bigg( f(u) + egin{cases} 0 & 	ext{若 } u 	ext{ 满足约束；} \ infty & 	ext{否则} end{cases} Bigg) 
onumber \
= & underset{u}{mathrm{min}} ; f(u), 	ext{ 且 } u 	ext{ 满足约束，}
end{align}
$$

[egin{align} & underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;; mathcal{L} (u, alpha, eta) onumber \ = & underset{u}{mathrm{min}} Bigg( f(u) + underset{alpha, eta}{mathrm{max}} Big( sumlimits_{i=1}^{m} alpha_i g_i (u) + sumlimits_{j=1}^{n} eta_j h_j (u) Big)Bigg) onumber \ = & underset{u}{mathrm{min}} Bigg( f(u) + egin{cases} 0 & ext{若 } u ext{ 满足约束；} \ infty & ext{否则} end{cases} Bigg) onumber \ = & underset{u}{mathrm{min}} ; f(u), ext{ 且 } u ext{ 满足约束，} end{align} ]

---

$$
egin{align}
underset{alpha, eta}{mathrm{max}} ;; underset{u}{mathrm{min}} ;;& mathcal{L} (u, alpha, eta) \
mathrm{s.t.} ;;;;;;& alpha_i geq 0, ;; i = 1,2,...,m. 
onumber
end{align}
$$

[egin{align} underset{alpha, eta}{mathrm{max}} ;; underset{u}{mathrm{min}} ;;& mathcal{L} (u, alpha, eta) \ mathrm{s.t.} ;;;;;;& alpha_i geq 0, ;; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
underset{alpha, eta}{mathrm{max}} ;; underset{u}{mathrm{min}} ;; mathcal{L} (u, alpha, eta) ;; leq ;; underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;; mathcal{L} (u, alpha, eta)
end{align}
$$

[egin{align} underset{alpha, eta}{mathrm{max}} ;; underset{u}{mathrm{min}} ;; mathcal{L} (u, alpha, eta) ;; leq ;; underset{u}{mathrm{min}} ;; underset{alpha, eta}{mathrm{max}} ;; mathcal{L} (u, alpha, eta) end{align} ]

---

$$
egin{align}
mathcal{L}(w,b,alpha) := frac{1}{2}w^{	op}w + sumlimits_{i=1}^{m}alpha_i ig(1- 
y_i (w^{	op} x_i + b) ig)
end{align}
$$

[egin{align} mathcal{L}(w,b,alpha) := frac{1}{2}w^{ op}w + sumlimits_{i=1}^{m}alpha_i ig(1- y_i (w^{ op} x_i + b) ig) end{align} ]

---

$$
egin{align}
underset{alpha}{mathrm{max}} ; underset{w,b}{mathrm{min}} ; & frac{1}{2}w^{	op}w + sumlimits_{i=1}^{m}alpha_i ig(1- y_i (w^{	op} x_i + b) ig) \
mathrm{s.t.} ;;;;;; & alpha_i geq 0, ; i = 1,2,...,m. 
onumber
end{align}
$$

[egin{align} underset{alpha}{mathrm{max}} ; underset{w,b}{mathrm{min}} ; & frac{1}{2}w^{ op}w + sumlimits_{i=1}^{m}alpha_i ig(1- y_i (w^{ op} x_i + b) ig) \ mathrm{s.t.} ;;;;;; & alpha_i geq 0, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
underset{alpha}{mathrm{min}} ;; & frac{1}{2} sumlimits_{i=1}^{m} sumlimits_{j=1}^{m} alpha_i alpha_j y_i y_j x_i^{	op} x_j - sumlimits_{i=1}^{m}alpha_i \
mathrm{s.t.} ;;; & sumlimits_{i=1}^{m} alpha_i y_i = 0, 
onumber \
& alpha_i geq 0, ; i = 1,2,...,m. 
onumber
end{align}
$$

[egin{align} underset{alpha}{mathrm{min}} ;; & frac{1}{2} sumlimits_{i=1}^{m} sumlimits_{j=1}^{m} alpha_i alpha_j y_i y_j x_i^{ op} x_j - sumlimits_{i=1}^{m}alpha_i \ mathrm{s.t.} ;;; & sumlimits_{i=1}^{m} alpha_i y_i = 0, onumber \ & alpha_i geq 0, ; i = 1,2,...,m. onumber end{align} ]

---

$$
egin{align}
frac{partial mathcal{L}}{partial w} = 0 Leftrightarrow & w = sumlimits_{i=1}^{m} alpha_i y_i x_i, \
frac{partial mathcal{L}}{partial b} = 0 Leftrightarrow & sumlimits_{i=1}^{m} alpha_i y_i.
end{align}
$$

[egin{align} frac{partial mathcal{L}}{partial w} = 0 Leftrightarrow & w = sumlimits_{i=1}^{m} alpha_i y_i x_i, \ frac{partial mathcal{L}}{partial b} = 0 Leftrightarrow & sumlimits_{i=1}^{m} alpha_i y_i. end{align} ]

---

$$
egin{align}
& u:=alpha,;mathcal{Q}:=[y_i y_j x_i^{	op} x_j]_{m 	imes m},;t:=-1,\
& c_i:=e_i,;d_i:=0,; i=1,2,...,m,\
& c_{m+1}:=[y_1;y_2; cdots ; y_m]^{	op} , ; d_{m+1}:=0,\
& c_{m+2}:=-[y_1;y_2; cdots ; y_m]^{	op}, ; d_{m+2}:=0,
end{align}
$$

[egin{align} & u:=alpha,;mathcal{Q}:=[y_i y_j x_i^{ op} x_j]_{m imes m},;t:=-1,\ & c_i:=e_i,;d_i:=0,; i=1,2,...,m,\ & c_{m+1}:=[y_1;y_2; cdots ; y_m]^{ op} , ; d_{m+1}:=0,\ & c_{m+2}:=-[y_1;y_2; cdots ; y_m]^{ op}, ; d_{m+2}:=0, end{align} ]

---

$$
egin{align}
u:=egin{bmatrix} w \ b end{bmatrix}, ;; g_i(u):= 1-y_i {egin{bmatrix} x_i \ 1 end{bmatrix}}^{	op} u,
end{align}
$$

[egin{align} u:=egin{bmatrix} w \ b end{bmatrix}, ;; g_i(u):= 1-y_i {egin{bmatrix} x_i \ 1 end{bmatrix}}^{ op} u, end{align} ]

---

$$
egin{align}
w = & sumlimits_{i=1}^{m}alpha_i y_i x_i 
onumber \
    = & sumlimits_{i:;alpha_i = 0}^{m} 0 cdot y_i x_i + sumlimits_{i:;alpha_i>0}^{m}alpha_i y_i x_i 
onumber \
    = &  sumlimits_{i in SV}^{}alpha_i y_i x_i,
end{align}
$$

[egin{align} w = & sumlimits_{i=1}^{m}alpha_i y_i x_i onumber \ = & sumlimits_{i:;alpha_i = 0}^{m} 0 cdot y_i x_i + sumlimits_{i:;alpha_i>0}^{m}alpha_i y_i x_i onumber \ = & sumlimits_{i in SV}^{}alpha_i y_i x_i, end{align} ]

---

$$
egin{align}
& y_s(w^{	op} x_s + b) = 1, 	ext{ 则} 
onumber \
& b = y_s - w^{	op} x_s = y_s - sumlimits_{i in SV} alpha_i y_i x_i^{	op} x_s.
end{align}
$$

[egin{align} & y_s(w^{ op} x_s + b) = 1, ext{ 则} onumber \ & b = y_s - w^{ op} x_s = y_s - sumlimits_{i in SV} alpha_i y_i x_i^{ op} x_s. end{align} ]

---

$$
egin{align}
h(x) = mathrm{sign} Big( sumlimits_{i in SV} alpha_i y_i x_i^{	op} x + b Big).
end{align}
$$

[egin{align} h(x) = mathrm{sign} Big( sumlimits_{i in SV} alpha_i y_i x_i^{ op} x + b Big). end{align} ]

---

$$
egin{align}
underset{w,b}{mathrm{min}} ;; & frac{1}{2} w^{	op} w \
mathrm{s.t.} ;; & y_i(w^{	op}phi(x_i) + b)geq1,;i=1,2,...,m; 
onumber \ 
onumber \ 
onumber \
underset{alpha}{mathrm{min}} ;; & frac{1}{2} sumlimits_{i=1}^{m}sumlimits_{j=1}^{m}alpha_ialpha_jy_iy_jphi(x_i)^{	op}phi(x_j)-sumlimits_{i=1}^{m}alpha_i \
mathrm{s.t.} ;; & sumlimits_{i=1}^{m}alpha_iy_i = 0, 
onumber \
&alpha_i geq 0, ; i=1,2,...,m. 
onumber
end{align}
$$

[egin{align} underset{w,b}{mathrm{min}} ;; & frac{1}{2} w^{ op} w \ mathrm{s.t.} ;; & y_i(w^{ op}phi(x_i) + b)geq1,;i=1,2,...,m; onumber \ onumber \ onumber \ underset{alpha}{mathrm{min}} ;; & frac{1}{2} sumlimits_{i=1}^{m}sumlimits_{j=1}^{m}alpha_ialpha_jy_iy_jphi(x_i)^{ op}phi(x_j)-sumlimits_{i=1}^{m}alpha_i \ mathrm{s.t.} ;; & sumlimits_{i=1}^{m}alpha_iy_i = 0, onumber \ &alpha_i geq 0, ; i=1,2,...,m. onumber end{align} ]

---

$egin{align}kappa(x_i, x_j)=phi (x_i)^T phi (x_j),end{align}$

(egin{align}kappa(x_i, x_j)=phi (x_i)^T phi (x_j),end{align})

---

$$
egin{align}
phi : x mapsto exp(-x^2) egin{bmatrix} 1\ sqrt{frac{2}{1}}x \ sqrt{frac{2^2}{2!}}x^2 \ vdots end{bmatrix}
end{align}
$$

[egin{align} phi : x mapsto exp(-x^2) egin{bmatrix} 1\ sqrt{frac{2}{1}}x \ sqrt{frac{2^2}{2!}}x^2 \ vdots end{bmatrix} end{align} ]

---

$$
egin{align}
kappa(x_i,x_j):=expBig(-(x_i - x_j)^2Big).
end{align}
$$

[egin{align} kappa(x_i,x_j):=expBig(-(x_i - x_j)^2Big). end{align} ]

---

$$
egin{align}
kappa(x_i,x_j) 
&= expBig(-(x_i - x_j)^2Big) 
onumber \
&= exp(-x_i^2)exp(-x_j^2)exp(2x_ix_j) 
onumber \
&= exp(-x_i^2)exp(-x_j^2)sumlimits_{k=0}^{infty}frac{(2x_ix_j)^k}{k!} 
onumber \
&= sumlimits_{k=0}^{infty}Bigg(exp(-x_i^2)sqrt{frac{2^k}{k!}}x_i^kBigg)Bigg(exp(-x_j^2)sqrt{frac{2^k}{k!}}x_j^kBigg) 
onumber \
&=  phi(x_i)^{	op}phi(x_j).
end{align}
$$

[egin{align} kappa(x_i,x_j) &= expBig(-(x_i - x_j)^2Big) onumber \ &= exp(-x_i^2)exp(-x_j^2)exp(2x_ix_j) onumber \ &= exp(-x_i^2)exp(-x_j^2)sumlimits_{k=0}^{infty}frac{(2x_ix_j)^k}{k!} onumber \ &= sumlimits_{k=0}^{infty}Bigg(exp(-x_i^2)sqrt{frac{2^k}{k!}}x_i^kBigg)Bigg(exp(-x_j^2)sqrt{frac{2^k}{k!}}x_j^kBigg) onumber \ &= phi(x_i)^{ op}phi(x_j). end{align} ]

---

$$
egin{align}
K := [kappa(x_i,x_j)]_{m 	imes m}
end{align}
$$

[egin{align} K := [kappa(x_i,x_j)]_{m imes m} end{align} ]

---

$$
egin{align}
Phi:=[phi(x_1);phi(x_2);ldots;phi(x_m)] in mathbb{R}^{	ilde{d} 	imes m},
end{align}
$$

[egin{align} Phi:=[phi(x_1);phi(x_2);ldots;phi(x_m)] in mathbb{R}^{ ilde{d} imes m}, end{align} ]

---

$$
egin{align}
c_1 kappa_1(x_i,x_j)+c_2 kappa_2 (x_i,x_j) = {egin{bmatrix} sqrt{c_1}phi_1(x_i) \ sqrt{c_2}phi_2(x_i) end{bmatrix}}^{	op} egin{bmatrix} sqrt{c_1}phi_1(x_i) \ sqrt{c_2}phi_2(x_i) end{bmatrix}\
kappa_1(x_i,x_j)kappa_2(x_i,x_j)=mathrm{vec}ig(phi_1(x_i)phi_2(x_i)^{	op}ig)^{	op}mathrm{vec}ig(phi_1(x_j)phi_2(x_j)^{	op}ig)^{	op},\
f(x_1)kappa_1(x_i,x_j)f(x_2)=ig(f(x_i)phi(x_i)^{	op}ig)^{	op}ig(f(x_j)phi(x_j)ig).
end{align}
$$

[egin{align} c_1 kappa_1(x_i,x_j)+c_2 kappa_2 (x_i,x_j) = {egin{bmatrix} sqrt{c_1}phi_1(x_i) \ sqrt{c_2}phi_2(x_i) end{bmatrix}}^{ op} egin{bmatrix} sqrt{c_1}phi_1(x_i) \ sqrt{c_2}phi_2(x_i) end{bmatrix}\ kappa_1(x_i,x_j)kappa_2(x_i,x_j)=mathrm{vec}ig(phi_1(x_i)phi_2(x_i)^{ op}ig)^{ op}mathrm{vec}ig(phi_1(x_j)phi_2(x_j)^{ op}ig)^{ op},\ f(x_1)kappa_1(x_i,x_j)f(x_2)=ig(f(x_i)phi(x_i)^{ op}ig)^{ op}ig(f(x_j)phi(x_j)ig). end{align} ]

---

$m+	ilde{d}+1$ 

(m+ ilde{d}+1)

---

$$
overbrace{
left[ egin{array}{c} ...W^{[1]T}{1}... ...W^{[1]T}{2}... ...W^{[1]T}{3}... ...W^{[1]T}{4}... end{array} 
ight]
}^{W^{[1]}}
*
overbrace{
left[ egin{array}{c} x_1 x_2 x_3 end{array} 
ight]
}^{input}
+
overbrace{
left[ egin{array}{c} b^{[1]}_1 b^{[1]}_2 b^{[1]}_3 b^{[1]}_4 end{array} 
ight]
}^{b^{[1]}}
$$

[overbrace{ left[ egin{array}{c} ...W^{[1]T}{1}... ...W^{[1]T}{2}... ...W^{[1]T}{3}... ...W^{[1]T}{4}... end{array} ight] }^{W^{[1]}} * overbrace{ left[ egin{array}{c} x_1 x_2 x_3 end{array} ight] }^{input} + overbrace{ left[ egin{array}{c} b^{[1]}_1 b^{[1]}_2 b^{[1]}_3 b^{[1]}_4 end{array} ight] }^{b^{[1]}} ]

---

$$
Z^{[1]}=
overbrace{
egin{bmatrix}
cdots w^{[1]T}_1 cdots \
cdots w^{[1]T}_2 cdots \
cdots w^{[1]T}_3 cdots \
cdots w^{[1]T}_4 cdots
end{bmatrix}
}^{W^{[1]},; (4 	imes 3)}
egin{bmatrix}
x_1 \
x_2 \
x_3
end{bmatrix}
+
overbrace{
egin{bmatrix}
b^{[1]}_1 \
b^{[1]}_2 \
b^{[1]}_3 \
b^{[1]}_4
end{bmatrix}
}^{b^{[1]},; (4 	imes 1)}
=
egin{bmatrix}
w^{[1]T}_1 x + b^{[1]}_1 \
w^{[1]T}_2 x + b^{[1]}_2 \
w^{[1]T}_3 x + b^{[1]}_3 \
w^{[1]T}_4 x + b^{[1]}_4 
end{bmatrix}
=
underbrace{
egin{bmatrix}
z^{[1]}_1 \
z^{[1]}_2 \
z^{[1]}_3 \
z^{[1]}_4
end{bmatrix}
}_{z^{[1]}}
$$

[Z^{[1]}= overbrace{ egin{bmatrix} cdots w^{[1]T}_1 cdots \ cdots w^{[1]T}_2 cdots \ cdots w^{[1]T}_3 cdots \ cdots w^{[1]T}_4 cdots end{bmatrix} }^{W^{[1]},; (4 imes 3)} egin{bmatrix} x_1 \ x_2 \ x_3 end{bmatrix} + overbrace{ egin{bmatrix} b^{[1]}_1 \ b^{[1]}_2 \ b^{[1]}_3 \ b^{[1]}_4 end{bmatrix} }^{b^{[1]},; (4 imes 1)} = egin{bmatrix} w^{[1]T}_1 x + b^{[1]}_1 \ w^{[1]T}_2 x + b^{[1]}_2 \ w^{[1]T}_3 x + b^{[1]}_3 \ w^{[1]T}_4 x + b^{[1]}_4 end{bmatrix} = underbrace{ egin{bmatrix} z^{[1]}_1 \ z^{[1]}_2 \ z^{[1]}_3 \ z^{[1]}_4 end{bmatrix} }_{z^{[1]}} ]

---