Markdown & LaTex 常用语法
2024-08-27 06:12:55
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> </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
$\mathrm{d}a$ : \(\mathrm{d}a\)
$da$ : \(da\)
$( \big( \Big( \bigg( \Bigg($ : \(( \big( \Big( \bigg( \Bigg(\)
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
- 嘻嘻哈哈
- 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>
- 有序列表1
- 有序列表2
- 有序列表3
1. 有序列表1
2. 有序列表2
3. 有序列表3
- 有序列表1
- 有序列表2
- 有序列表3
1. 有序列表1
1. 有序列表2
1. 有序列表3
- 有序列表1
- 有序列表2
- 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
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 \tag{n} 为公式添加编号
不使用 \begin{align} 和 \end{align} 也可以为公式添加标号，可以使用 \tag{n}
$aaa \tag{1}$
$bbb \tag{2}$
\(aaa \tag{1}\)
\(bbb \tag{2}\)
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="width: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
$$
\begin{align}
& \underset{w,b}{\mathrm{max}} \;\; \underset{i}{\mathrm{min}} \;\; \frac{2}{||w||} | w^{\top} x_i + b |, \\
& \mathrm{s.t.} \;\;y_i(w^{\top}x_i + b) > 0, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
& \underset{w,b}{\mathrm{max}} \;\; \underset{i}{\mathrm{min}} \;\; \frac{2}{||w||} | w^{\top} x_i + b |, \\
& \mathrm{s.t.} \;\;y_i(w^{\top}x_i + b) > 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
& \underset{w,b}{\mathrm{max}} \;\; \underset{i}{\mathrm{min}} \;\; \frac{2}{||w||} | w^{\top} x_i + b |, \\
& \mathrm{s.t.} \;\;y_i(w^{\top}x_i + b) > 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
h(x_i)=
\begin{cases}
1 & \text{若 }y_i=1; \\
-1 & \text{若 }y_i=-1. \\
\end{cases}
\end{align}
$$
\(\begin{align}h(x_i)=\begin{cases} 1 & \text{若 }y_i=1; \\ -1 & \text{若 }y_i=-1. \\\end{cases}\end{align}\)
$$
\begin{align}
h(x_i) := \mathrm{sign}(w^T x_i + b), \text{ 其中 } w_i \in \mathbb{R}^d,b \in \mathbb{R}.
\end{align}
$$
\(\begin{align}h(x_i) := \mathrm{sign}(w^T x_i + b), \text{ 其中 } w_i \in \mathbb{R}^d,b \in \mathbb{R}.\end{align}\)
$$
\begin{align}
\forall_{i.} \;\; y_i(w^T x_i + b) > 0
\end{align}
$$
\(\begin{align} \forall_{i.} \;\; y_i(w^T x_i + b) > 0 \end{align}\)
$$
\begin{align}
y_i h(x_i) = 1 \Leftrightarrow y_i \mathrm{sign}(w^T x_i + b) = 1 \Leftrightarrow y_i(w^T x_i + b) > 0
\end{align}
$$
\(\begin{align} y_i h(x_i) = 1 \Leftrightarrow y_i \mathrm{sign}(w^T x_i + b) = 1 \Leftrightarrow y_i(w^T x_i + b) > 0 \end{align}\)
$$
\begin{align}
\frac{1}{||w||} | w^{\top} p + b |
\end{align}
$$
\(\begin{align} \frac{1}{||w||} | w^{\top} p + b |\end{align}\)
$$
\begin{align}
w^{\top}(x_1-x_2) = w^{\top}x_1-w^{\top}x_2=(-b)-(-b)=0,
\end{align}
$$
\(\begin{align} w^{\top}(x_1-x_2) = w^{\top}x_1-w^{\top}x_2=(-b)-(-b)=0,\end{align}\)
\(w \perp (x_1 - x_2)\)
$$
\begin{align}
\mathrm{proj}_w(p-x)
&= ||p-x|| \cdot |\cos (w, p - x)| \nonumber\\
&= ||p-x|| \cdot \frac{|w^{\top}(p-x)|}{||w|| \cdot ||p-x||} \nonumber \\
&= \frac{1}{||w||} |w^{\top}p - w^{\top}x| \nonumber \\
&= \frac{1}{||w||} | w^{\top}p + b |
\end{align}
$$
\(\begin{align} \mathrm{proj}_w(p-x) &= ||p-x|| \cdot |\cos (w, p - x)| \nonumber\\ &= ||p-x|| \cdot \frac{|w^{\top}(p-x)|}{||w|| \cdot ||p-x||} \nonumber \\ &= \frac{1}{||w||} |w^{\top}p - w^{\top}x| \nonumber \\ &= \frac{1}{||w||} | w^{\top}p + b | \end{align}\)
$$
\gamma := 2 \; \underset{i}{\mathrm{min}} \frac{1}{||w||} | w^{\top} x_i + b |
$$
\(\gamma := 2 \; \underset{i}{\mathrm{min}} \frac{1}{||w||} | w^{\top} x_i + b |\)
$$
\begin{align}
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
\]
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
\]
& \underset{u}{\mathrm{min}} \;\; \frac{1}{2} u^{\top}Qu+t^{\top}u \\
& \mathrm{s.t.} \;\; c_i^{\top} u \geq d_i, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
\frac{2}{||rw^*||} | (rw^*)^{\top} x_i + rb^* | = \frac{2}{||w^*||} | w^{*\top} x_i + b^* |, \\
y_i \big( (rw^*)^{\top} x_i + rb^* \big) > 0 \Leftrightarrow y_i (w^{*\top} x_i + b^*)>0.
\end{align}
$$
\[\begin{align}
\frac{2}{||rw^*||} | (rw^*)^{\top} x_i + rb^* | = \frac{2}{||w^*||} | w^{*\top} x_i + b^* |, \\
y_i \big( (rw^*)^{\top} x_i + rb^* \big) > 0 \Leftrightarrow y_i (w^{*\top} x_i + b^*)>0.
\end{align}
\]
\frac{2}{||rw^*||} | (rw^*)^{\top} x_i + rb^* | = \frac{2}{||w^*||} | w^{*\top} x_i + b^* |, \\
y_i \big( (rw^*)^{\top} x_i + rb^* \big) > 0 \Leftrightarrow y_i (w^{*\top} x_i + b^*)>0.
\end{align}
\]
$$
\begin{align}
\underset{i}{\mathrm{min}} \; | w^{\top} x_i + b | = 1.
\end{align}
$$
\[\begin{align}
\underset{i}{\mathrm{min}} \; | w^{\top} x_i + b | = 1.
\end{align}
\]
\underset{i}{\mathrm{min}} \; | w^{\top} x_i + b | = 1.
\end{align}
\]
$$
\begin{align}
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; y_i (w^{\top} x_i +b ) \geq 1, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; y_i (w^{\top} x_i +b ) \geq 1, \; i = 1,2,...,m. \nonumber
\end{align}
\]
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; y_i (w^{\top} x_i +b ) \geq 1, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; \underset{i}{\mathrm{min}} \;\; y_i (w^{\top} x_i +b ) = 1. \nonumber
\end{align}
$$
\[\begin{align}
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; \underset{i}{\mathrm{min}} \;\; y_i (w^{\top} x_i +b ) = 1. \nonumber
\end{align}
\]
& \underset{w, b}{\mathrm{min}} \;\; \frac{1}{2} w^{\top} w \\
& \mathrm{s.t.} \;\; \underset{i}{\mathrm{min}} \;\; y_i (w^{\top} x_i +b ) = 1. \nonumber
\end{align}
\]
$$
\begin{align}
\underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} w^{\top} w
& = \underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} ||w|| \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \frac{2}{||w||} \cdot 1 \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} y_i (w^{\top} x_i + b) \Big) \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} |w^{\top} x_i + b| \Big)
\end{align}
$$
\[\begin{align}
\underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} w^{\top} w
& = \underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} ||w|| \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \frac{2}{||w||} \cdot 1 \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} y_i (w^{\top} x_i + b) \Big) \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} |w^{\top} x_i + b| \Big)
\end{align}
\]
\underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} w^{\top} w
& = \underset{w, b}{\mathrm{argmin}} \;\; \frac{1}{2} ||w|| \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \frac{2}{||w||} \cdot 1 \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} y_i (w^{\top} x_i + b) \Big) \nonumber\\
& = \underset{w, b}{\mathrm{argmax}} \;\; \Big( \underset{i}{\mathrm{min}} \; \frac{2}{||w||} |w^{\top} x_i + b| \Big)
\end{align}
\]
$$
\begin{align}
u := \begin{bmatrix} w \\ b \end{bmatrix}, Q := \begin{bmatrix} I&0\\0&0 \end{bmatrix}, t := 0, \\
c_i := y_i \begin{bmatrix} x_i \\ 1 \end{bmatrix}, d_i := 1,
\end{align}
$$
\[\begin{align}
u := \begin{bmatrix} w \\ b \end{bmatrix}, Q := \begin{bmatrix} I&0\\0&0 \end{bmatrix}, t := 0, \\
c_i := y_i \begin{bmatrix} x_i \\ 1 \end{bmatrix}, d_i := 1,
\end{align}
\]
u := \begin{bmatrix} w \\ b \end{bmatrix}, Q := \begin{bmatrix} I&0\\0&0 \end{bmatrix}, t := 0, \\
c_i := y_i \begin{bmatrix} x_i \\ 1 \end{bmatrix}, d_i := 1,
\end{align}
\]
$$
\begin{align}
\underset{u}{\mathrm{min}} &\;\; f(u) &\\
\mathrm{s.t.} &\;\; g_i (u) \leq 0, &i = 1,2,...,m, \nonumber\\
& \;\; h_j (u) = 0, &j = 1,2,...,n, \nonumber
\end{align}
$$
\[\begin{align}
\underset{u}{\mathrm{min}} &\;\; f(u) &\\
\mathrm{s.t.} &\;\; g_i (u) \leq 0, &i = 1,2,...,m, \nonumber\\
& \;\; h_j (u) = 0, &j = 1,2,...,n, \nonumber
\end{align}
\]
\underset{u}{\mathrm{min}} &\;\; f(u) &\\
\mathrm{s.t.} &\;\; g_i (u) \leq 0, &i = 1,2,...,m, \nonumber\\
& \;\; h_j (u) = 0, &j = 1,2,...,n, \nonumber
\end{align}
\]
$$
\begin{align}
\mathcal{L}(u,\alpha,\beta) := f(u) + \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u)
\end{align}
$$
\[\begin{align}
\mathcal{L}(u,\alpha,\beta) := f(u) + \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u)
\end{align}
\]
\mathcal{L}(u,\alpha,\beta) := f(u) + \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u)
\end{align}
\]
$$
\begin{align}
\underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
\underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
\]
\underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
& \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \underset{\alpha, \beta}{\mathrm{max}} \Big( \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u) \Big)\Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \begin{cases} 0 & \text{若 } u \text{ 满足约束；} \\ \infty & \text{否则} \end{cases} \Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \; f(u), \text{ 且 } u \text{ 满足约束，}
\end{align}
$$
\[\begin{align}
& \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \underset{\alpha, \beta}{\mathrm{max}} \Big( \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u) \Big)\Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \begin{cases} 0 & \text{若 } u \text{ 满足约束；} \\ \infty & \text{否则} \end{cases} \Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \; f(u), \text{ 且 } u \text{ 满足约束，}
\end{align}
\]
& \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \underset{\alpha, \beta}{\mathrm{max}} \Big( \sum\limits_{i=1}^{m} \alpha_i g_i (u) + \sum\limits_{j=1}^{n} \beta_j h_j (u) \Big)\Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \Bigg( f(u) + \begin{cases} 0 & \text{若 } u \text{ 满足约束；} \\ \infty & \text{否则} \end{cases} \Bigg) \nonumber \\
= & \underset{u}{\mathrm{min}} \; f(u), \text{ 且 } u \text{ 满足约束，}
\end{align}
\]
$$
\begin{align}
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
\]
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\;& \mathcal{L} (u, \alpha, \beta) \\
\mathrm{s.t.} \;\;\;\;\;\;& \alpha_i \geq 0, \;\; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\; \mathcal{L} (u, \alpha, \beta) \;\; \leq \;\; \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta)
\end{align}
$$
\[\begin{align}
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\; \mathcal{L} (u, \alpha, \beta) \;\; \leq \;\; \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta)
\end{align}
\]
\underset{\alpha, \beta}{\mathrm{max}} \;\; \underset{u}{\mathrm{min}} \;\; \mathcal{L} (u, \alpha, \beta) \;\; \leq \;\; \underset{u}{\mathrm{min}} \;\; \underset{\alpha, \beta}{\mathrm{max}} \;\; \mathcal{L} (u, \alpha, \beta)
\end{align}
\]
$$
\begin{align}
\mathcal{L}(w,b,\alpha) := \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1-
y_i (w^{\top} x_i + b) \big)
\end{align}
$$
\[\begin{align}
\mathcal{L}(w,b,\alpha) := \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1-
y_i (w^{\top} x_i + b) \big)
\end{align}
\]
\mathcal{L}(w,b,\alpha) := \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1-
y_i (w^{\top} x_i + b) \big)
\end{align}
\]
$$
\begin{align}
\underset{\alpha}{\mathrm{max}} \; \underset{w,b}{\mathrm{min}} \; & \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1- y_i (w^{\top} x_i + b) \big) \\
\mathrm{s.t.} \;\;\;\;\;\; & \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
\underset{\alpha}{\mathrm{max}} \; \underset{w,b}{\mathrm{min}} \; & \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1- y_i (w^{\top} x_i + b) \big) \\
\mathrm{s.t.} \;\;\;\;\;\; & \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
\underset{\alpha}{\mathrm{max}} \; \underset{w,b}{\mathrm{min}} \; & \frac{1}{2}w^{\top}w + \sum\limits_{i=1}^{m}\alpha_i \big(1- y_i (w^{\top} x_i + b) \big) \\
\mathrm{s.t.} \;\;\;\;\;\; & \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m} \sum\limits_{j=1}^{m} \alpha_i \alpha_j y_i y_j x_i^{\top} x_j - \sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\;\; & \sum\limits_{i=1}^{m} \alpha_i y_i = 0, \nonumber \\
& \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m} \sum\limits_{j=1}^{m} \alpha_i \alpha_j y_i y_j x_i^{\top} x_j - \sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\;\; & \sum\limits_{i=1}^{m} \alpha_i y_i = 0, \nonumber \\
& \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m} \sum\limits_{j=1}^{m} \alpha_i \alpha_j y_i y_j x_i^{\top} x_j - \sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\;\; & \sum\limits_{i=1}^{m} \alpha_i y_i = 0, \nonumber \\
& \alpha_i \geq 0, \; i = 1,2,...,m. \nonumber
\end{align}
\]
$$
\begin{align}
\frac{\partial \mathcal{L}}{\partial w} = 0 \Leftrightarrow & w = \sum\limits_{i=1}^{m} \alpha_i y_i x_i, \\
\frac{\partial \mathcal{L}}{\partial b} = 0 \Leftrightarrow & \sum\limits_{i=1}^{m} \alpha_i y_i.
\end{align}
$$
\[\begin{align}
\frac{\partial \mathcal{L}}{\partial w} = 0 \Leftrightarrow & w = \sum\limits_{i=1}^{m} \alpha_i y_i x_i, \\
\frac{\partial \mathcal{L}}{\partial b} = 0 \Leftrightarrow & \sum\limits_{i=1}^{m} \alpha_i y_i.
\end{align}
\]
\frac{\partial \mathcal{L}}{\partial w} = 0 \Leftrightarrow & w = \sum\limits_{i=1}^{m} \alpha_i y_i x_i, \\
\frac{\partial \mathcal{L}}{\partial b} = 0 \Leftrightarrow & \sum\limits_{i=1}^{m} \alpha_i y_i.
\end{align}
\]
$$
\begin{align}
& u:=\alpha,\;\mathcal{Q}:=[y_i y_j x_i^{\top} x_j]_{m \times m},\;t:=-1,\\
& c_i:=e_i,\;d_i:=0,\; i=1,2,...,m,\\
& c_{m+1}:=[y_1\;y_2\; \cdots \; y_m]^{\top} , \; d_{m+1}:=0,\\
& c_{m+2}:=-[y_1\;y_2\; \cdots \; y_m]^{\top}, \; d_{m+2}:=0,
\end{align}
$$
\[\begin{align}
& u:=\alpha,\;\mathcal{Q}:=[y_i y_j x_i^{\top} x_j]_{m \times m},\;t:=-1,\\
& c_i:=e_i,\;d_i:=0,\; i=1,2,...,m,\\
& c_{m+1}:=[y_1\;y_2\; \cdots \; y_m]^{\top} , \; d_{m+1}:=0,\\
& c_{m+2}:=-[y_1\;y_2\; \cdots \; y_m]^{\top}, \; d_{m+2}:=0,
\end{align}
\]
& u:=\alpha,\;\mathcal{Q}:=[y_i y_j x_i^{\top} x_j]_{m \times m},\;t:=-1,\\
& c_i:=e_i,\;d_i:=0,\; i=1,2,...,m,\\
& c_{m+1}:=[y_1\;y_2\; \cdots \; y_m]^{\top} , \; d_{m+1}:=0,\\
& c_{m+2}:=-[y_1\;y_2\; \cdots \; y_m]^{\top}, \; d_{m+2}:=0,
\end{align}
\]
$$
\begin{align}
u:=\begin{bmatrix} w \\ b \end{bmatrix}, \;\; g_i(u):= 1-y_i {\begin{bmatrix} x_i \\ 1 \end{bmatrix}}^{\top} u,
\end{align}
$$
\[\begin{align}
u:=\begin{bmatrix} w \\ b \end{bmatrix}, \;\; g_i(u):= 1-y_i {\begin{bmatrix} x_i \\ 1 \end{bmatrix}}^{\top} u,
\end{align}
\]
u:=\begin{bmatrix} w \\ b \end{bmatrix}, \;\; g_i(u):= 1-y_i {\begin{bmatrix} x_i \\ 1 \end{bmatrix}}^{\top} u,
\end{align}
\]
$$
\begin{align}
w = & \sum\limits_{i=1}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i:\;\alpha_i = 0}^{m} 0 \cdot y_i x_i + \sum\limits_{i:\;\alpha_i>0}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i \in SV}^{}\alpha_i y_i x_i,
\end{align}
$$
\[\begin{align}
w = & \sum\limits_{i=1}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i:\;\alpha_i = 0}^{m} 0 \cdot y_i x_i + \sum\limits_{i:\;\alpha_i>0}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i \in SV}^{}\alpha_i y_i x_i,
\end{align}
\]
w = & \sum\limits_{i=1}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i:\;\alpha_i = 0}^{m} 0 \cdot y_i x_i + \sum\limits_{i:\;\alpha_i>0}^{m}\alpha_i y_i x_i \nonumber \\
= & \sum\limits_{i \in SV}^{}\alpha_i y_i x_i,
\end{align}
\]
$$
\begin{align}
& y_s(w^{\top} x_s + b) = 1, \text{ 则} \nonumber \\
& b = y_s - w^{\top} x_s = y_s - \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x_s.
\end{align}
$$
\[\begin{align}
& y_s(w^{\top} x_s + b) = 1, \text{ 则} \nonumber \\
& b = y_s - w^{\top} x_s = y_s - \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x_s.
\end{align}
\]
& y_s(w^{\top} x_s + b) = 1, \text{ 则} \nonumber \\
& b = y_s - w^{\top} x_s = y_s - \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x_s.
\end{align}
\]
$$
\begin{align}
h(x) = \mathrm{sign} \Big( \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x + b \Big).
\end{align}
$$
\[\begin{align}
h(x) = \mathrm{sign} \Big( \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x + b \Big).
\end{align}
\]
h(x) = \mathrm{sign} \Big( \sum\limits_{i \in SV} \alpha_i y_i x_i^{\top} x + b \Big).
\end{align}
\]
$$
\begin{align}
\underset{w,b}{\mathrm{min}} \;\; & \frac{1}{2} w^{\top} w \\
\mathrm{s.t.} \;\; & y_i(w^{\top}\phi(x_i) + b)\geq1,\;i=1,2,...,m; \nonumber \\ \nonumber \\ \nonumber \\
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\alpha_i\alpha_jy_iy_j\phi(x_i)^{\top}\phi(x_j)-\sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\; & \sum\limits_{i=1}^{m}\alpha_iy_i = 0, \nonumber \\
&\alpha_i \geq 0, \; i=1,2,...,m. \nonumber
\end{align}
$$
\[\begin{align}
\underset{w,b}{\mathrm{min}} \;\; & \frac{1}{2} w^{\top} w \\
\mathrm{s.t.} \;\; & y_i(w^{\top}\phi(x_i) + b)\geq1,\;i=1,2,...,m; \nonumber \\ \nonumber \\ \nonumber \\
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\alpha_i\alpha_jy_iy_j\phi(x_i)^{\top}\phi(x_j)-\sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\; & \sum\limits_{i=1}^{m}\alpha_iy_i = 0, \nonumber \\
&\alpha_i \geq 0, \; i=1,2,...,m. \nonumber
\end{align}
\]
\underset{w,b}{\mathrm{min}} \;\; & \frac{1}{2} w^{\top} w \\
\mathrm{s.t.} \;\; & y_i(w^{\top}\phi(x_i) + b)\geq1,\;i=1,2,...,m; \nonumber \\ \nonumber \\ \nonumber \\
\underset{\alpha}{\mathrm{min}} \;\; & \frac{1}{2} \sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\alpha_i\alpha_jy_iy_j\phi(x_i)^{\top}\phi(x_j)-\sum\limits_{i=1}^{m}\alpha_i \\
\mathrm{s.t.} \;\; & \sum\limits_{i=1}^{m}\alpha_iy_i = 0, \nonumber \\
&\alpha_i \geq 0, \; i=1,2,...,m. \nonumber
\end{align}
\]
$\begin{align}\kappa(x_i, x_j)=\phi (x_i)^T \phi (x_j),\end{align}$
\(\begin{align}\kappa(x_i, x_j)=\phi (x_i)^T \phi (x_j),\end{align}\)
$$
\begin{align}
\phi : x \mapsto exp(-x^2) \begin{bmatrix} 1\\ \sqrt{\frac{2}{1}}x \\ \sqrt{\frac{2^2}{2!}}x^2 \\ \vdots \end{bmatrix}
\end{align}
$$
\[\begin{align}
\phi : x \mapsto exp(-x^2) \begin{bmatrix} 1\\ \sqrt{\frac{2}{1}}x \\ \sqrt{\frac{2^2}{2!}}x^2 \\ \vdots \end{bmatrix}
\end{align}
\]
\phi : x \mapsto exp(-x^2) \begin{bmatrix} 1\\ \sqrt{\frac{2}{1}}x \\ \sqrt{\frac{2^2}{2!}}x^2 \\ \vdots \end{bmatrix}
\end{align}
\]
$$
\begin{align}
\kappa(x_i,x_j):=exp\Big(-(x_i - x_j)^2\Big).
\end{align}
$$
\[\begin{align}
\kappa(x_i,x_j):=exp\Big(-(x_i - x_j)^2\Big).
\end{align}
\]
\kappa(x_i,x_j):=exp\Big(-(x_i - x_j)^2\Big).
\end{align}
\]
$$
\begin{align}
\kappa(x_i,x_j)
&= exp\Big(-(x_i - x_j)^2\Big) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)exp(2x_ix_j) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)\sum\limits_{k=0}^{\infty}\frac{(2x_ix_j)^k}{k!} \nonumber \\
&= \sum\limits_{k=0}^{\infty}\Bigg(exp(-x_i^2)\sqrt{\frac{2^k}{k!}}x_i^k\Bigg)\Bigg(exp(-x_j^2)\sqrt{\frac{2^k}{k!}}x_j^k\Bigg) \nonumber \\
&= \phi(x_i)^{\top}\phi(x_j).
\end{align}
$$
\[\begin{align}
\kappa(x_i,x_j)
&= exp\Big(-(x_i - x_j)^2\Big) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)exp(2x_ix_j) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)\sum\limits_{k=0}^{\infty}\frac{(2x_ix_j)^k}{k!} \nonumber \\
&= \sum\limits_{k=0}^{\infty}\Bigg(exp(-x_i^2)\sqrt{\frac{2^k}{k!}}x_i^k\Bigg)\Bigg(exp(-x_j^2)\sqrt{\frac{2^k}{k!}}x_j^k\Bigg) \nonumber \\
&= \phi(x_i)^{\top}\phi(x_j).
\end{align}
\]
\kappa(x_i,x_j)
&= exp\Big(-(x_i - x_j)^2\Big) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)exp(2x_ix_j) \nonumber \\
&= exp(-x_i^2)exp(-x_j^2)\sum\limits_{k=0}^{\infty}\frac{(2x_ix_j)^k}{k!} \nonumber \\
&= \sum\limits_{k=0}^{\infty}\Bigg(exp(-x_i^2)\sqrt{\frac{2^k}{k!}}x_i^k\Bigg)\Bigg(exp(-x_j^2)\sqrt{\frac{2^k}{k!}}x_j^k\Bigg) \nonumber \\
&= \phi(x_i)^{\top}\phi(x_j).
\end{align}
\]
$$
\begin{align}
K := [\kappa(x_i,x_j)]_{m \times m}
\end{align}
$$
\[\begin{align}
K := [\kappa(x_i,x_j)]_{m \times m}
\end{align}
\]
K := [\kappa(x_i,x_j)]_{m \times m}
\end{align}
\]
$$
\begin{align}
\Phi:=[\phi(x_1)\;\phi(x_2)\;\ldots\;\phi(x_m)] \in \mathbb{R}^{\tilde{d} \times m},
\end{align}
$$
\[\begin{align}
\Phi:=[\phi(x_1)\;\phi(x_2)\;\ldots\;\phi(x_m)] \in \mathbb{R}^{\tilde{d} \times m},
\end{align}
\]
\Phi:=[\phi(x_1)\;\phi(x_2)\;\ldots\;\phi(x_m)] \in \mathbb{R}^{\tilde{d} \times m},
\end{align}
\]
$$
\begin{align}
c_1 \kappa_1(x_i,x_j)+c_2 \kappa_2 (x_i,x_j) = {\begin{bmatrix} \sqrt{c_1}\phi_1(x_i) \\ \sqrt{c_2}\phi_2(x_i) \end{bmatrix}}^{\top} \begin{bmatrix} \sqrt{c_1}\phi_1(x_i) \\ \sqrt{c_2}\phi_2(x_i) \end{bmatrix}\\
\kappa_1(x_i,x_j)\kappa_2(x_i,x_j)=\mathrm{vec}\big(\phi_1(x_i)\phi_2(x_i)^{\top}\big)^{\top}\mathrm{vec}\big(\phi_1(x_j)\phi_2(x_j)^{\top}\big)^{\top},\\
f(x_1)\kappa_1(x_i,x_j)f(x_2)=\big(f(x_i)\phi(x_i)^{\top}\big)^{\top}\big(f(x_j)\phi(x_j)\big).
\end{align}
$$
\[\begin{align}
c_1 \kappa_1(x_i,x_j)+c_2 \kappa_2 (x_i,x_j) = {\begin{bmatrix} \sqrt{c_1}\phi_1(x_i) \\ \sqrt{c_2}\phi_2(x_i) \end{bmatrix}}^{\top} \begin{bmatrix} \sqrt{c_1}\phi_1(x_i) \\ \sqrt{c_2}\phi_2(x_i) \end{bmatrix}\\
\kappa_1(x_i,x_j)\kappa_2(x_i,x_j)=\mathrm{vec}\big(\phi_1(x_i)\phi_2(x_i)^{\top}\big)^{\top}\mathrm{vec}\big(\phi_1(x_j)\phi_2(x_j)^{\top}\big)^{\top},\\
f(x_1)\kappa_1(x_i,x_j)f(x_2)=\big(f(x_i)\phi(x_i)^{\top}\big)^{\top}\big(f(x_j)\phi(x_j)\big).
\end{align}
\]
c_1 \kappa_1(x_i,x_j)+c_2 \kappa_2 (x_i,x_j) = {\begin{bmatrix} \sqrt{c_1}\phi_1(x_i) \\ \sqrt{c_2}\phi_2(x_i) \end{bmatrix}}^{\top} \begin{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}\big(\phi_1(x_i)\phi_2(x_i)^{\top}\big)^{\top}\mathrm{vec}\big(\phi_1(x_j)\phi_2(x_j)^{\top}\big)^{\top},\\
f(x_1)\kappa_1(x_i,x_j)f(x_2)=\big(f(x_i)\phi(x_i)^{\top}\big)^{\top}\big(f(x_j)\phi(x_j)\big).
\end{align}
\]
$m+\tilde{d}+1$
\(m+\tilde{d}+1\)
$$
\overbrace{
\left[ \begin{array}{c} ...W^{[1]T}{1}...\ ...W^{[1]T}{2}...\ ...W^{[1]T}{3}...\ ...W^{[1]T}{4}... \end{array} \right]
}^{W^{[1]}}
*
\overbrace{
\left[ \begin{array}{c} x_1\ x_2\ x_3\ \end{array} \right]
}^{input}
+
\overbrace{
\left[ \begin{array}{c} b^{[1]}_1\ b^{[1]}_2\ b^{[1]}_3\ b^{[1]}_4\ \end{array} \right]
}^{b^{[1]}}
$$
\[\overbrace{
\left[ \begin{array}{c} ...W^{[1]T}{1}...\ ...W^{[1]T}{2}...\ ...W^{[1]T}{3}...\ ...W^{[1]T}{4}... \end{array} \right]
}^{W^{[1]}}
*
\overbrace{
\left[ \begin{array}{c} x_1\ x_2\ x_3\ \end{array} \right]
}^{input}
+
\overbrace{
\left[ \begin{array}{c} b^{[1]}_1\ b^{[1]}_2\ b^{[1]}_3\ b^{[1]}_4\ \end{array} \right]
}^{b^{[1]}}
\]
\left[ \begin{array}{c} ...W^{[1]T}{1}...\ ...W^{[1]T}{2}...\ ...W^{[1]T}{3}...\ ...W^{[1]T}{4}... \end{array} \right]
}^{W^{[1]}}
*
\overbrace{
\left[ \begin{array}{c} x_1\ x_2\ x_3\ \end{array} \right]
}^{input}
+
\overbrace{
\left[ \begin{array}{c} b^{[1]}_1\ b^{[1]}_2\ b^{[1]}_3\ b^{[1]}_4\ \end{array} \right]
}^{b^{[1]}}
\]
$$
Z^{[1]}=
\overbrace{
\begin{bmatrix}
\cdots w^{[1]T}_1 \cdots \\
\cdots w^{[1]T}_2 \cdots \\
\cdots w^{[1]T}_3 \cdots \\
\cdots w^{[1]T}_4 \cdots
\end{bmatrix}
}^{W^{[1]},\; (4 \times 3)}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
+
\overbrace{
\begin{bmatrix}
b^{[1]}_1 \\
b^{[1]}_2 \\
b^{[1]}_3 \\
b^{[1]}_4
\end{bmatrix}
}^{b^{[1]},\; (4 \times 1)}
=
\begin{bmatrix}
w^{[1]T}_1 x + b^{[1]}_1 \\
w^{[1]T}_2 x + b^{[1]}_2 \\
w^{[1]T}_3 x + b^{[1]}_3 \\
w^{[1]T}_4 x + b^{[1]}_4
\end{bmatrix}
=
\underbrace{
\begin{bmatrix}
z^{[1]}_1 \\
z^{[1]}_2 \\
z^{[1]}_3 \\
z^{[1]}_4
\end{bmatrix}
}_{z^{[1]}}
$$
\[Z^{[1]}=
\overbrace{
\begin{bmatrix}
\cdots w^{[1]T}_1 \cdots \\
\cdots w^{[1]T}_2 \cdots \\
\cdots w^{[1]T}_3 \cdots \\
\cdots w^{[1]T}_4 \cdots
\end{bmatrix}
}^{W^{[1]},\; (4 \times 3)}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
+
\overbrace{
\begin{bmatrix}
b^{[1]}_1 \\
b^{[1]}_2 \\
b^{[1]}_3 \\
b^{[1]}_4
\end{bmatrix}
}^{b^{[1]},\; (4 \times 1)}
=
\begin{bmatrix}
w^{[1]T}_1 x + b^{[1]}_1 \\
w^{[1]T}_2 x + b^{[1]}_2 \\
w^{[1]T}_3 x + b^{[1]}_3 \\
w^{[1]T}_4 x + b^{[1]}_4
\end{bmatrix}
=
\underbrace{
\begin{bmatrix}
z^{[1]}_1 \\
z^{[1]}_2 \\
z^{[1]}_3 \\
z^{[1]}_4
\end{bmatrix}
}_{z^{[1]}}
\]
\overbrace{
\begin{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 \times 3)}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
+
\overbrace{
\begin{bmatrix}
b^{[1]}_1 \\
b^{[1]}_2 \\
b^{[1]}_3 \\
b^{[1]}_4
\end{bmatrix}
}^{b^{[1]},\; (4 \times 1)}
=
\begin{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{
\begin{bmatrix}
z^{[1]}_1 \\
z^{[1]}_2 \\
z^{[1]}_3 \\
z^{[1]}_4
\end{bmatrix}
}_{z^{[1]}}
\]
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