向量与矩阵的范数（比较1-范数、2-范数、无穷范数、p-范数、L0范数和 L1范数等）

男娘i 2022-05-16 02:50 378阅读 0赞

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阅读文献时，经常看到各种范数，机器学习中的稀疏模型等，也有各种范数，其名称往往容易混淆，例如：L1范数也常称为“1-范数”，但又和真正的1-范数又有很大区别。下面将依次介绍各种范数。

## **1、向量的范数** ##

**向量的1-范数： ![\{\\left\\| X \\right\\|\_1\} = \\sum\\limits\_\{i = 1\}^n \{\\left| \{\{x\_i\}\} \\right|\}][left_ X _right_1_ _ _sum_limits_i _ 1_n _left_ _x_i_ _right] ;** 各个元素的绝对值之和；

**向量的2-范数：![\{\\left\\| X \\right\\|\_2\} = \{\\left( \{\\sum\\limits\_\{i = 1\}^n \{\{x\_i\}^2\} \} \\right)^\{\\frac\{1\}\{2\}\}\} = \\sqrt \{\\sum\\limits\_\{i = 1\}^n \{\{x\_i\}^2\} \}][left_ X _right_2_ _ _left_ _sum_limits_i _ 1_n _x_i_2_ _ _right_frac_1_2_ _ _sqrt _sum_limits_i _ 1_n _x_i_2_]**；每个元素的平方和再开平方根；

**向量的无穷范数：**![\{\\left\\| X \\right\\|\_\\infty \} = \\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{x\_i\}\} \\right|][left_ X _right_infty _ _ _mathop _max _limits_1 _le i _le n_ _left_ _x_i_ _right]

***p*\-范数：**![\{\\left\\| X \\right\\|\_p\} = \{\\left( \{\\sum\\limits\_\{i = 1\}^n \{\{\{\\left| \{\{x\_i\}\} \\right|\}^p\}\} \} \\right)^\{\\frac\{1\}\{p\}\}\}][left_ X _right_p_ _ _left_ _sum_limits_i _ 1_n _left_ _x_i_ _right_p_ _ _right_frac_1_p]，其中正整数*p*≥1，并且有![\\mathop \{\\lim \}\\limits\_\{p \\to \\infty \} \{\\left\\| X \\right\\|\_p\} = \\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{x\_i\}\} \\right|][mathop _lim _limits_p _to _infty _ _left_ X _right_p_ _ _mathop _max _limits_1 _le i _le n_ _left_ _x_i_ _right]

例：向量*X*=\[2, 3, -5, -7\] ，求向量的1-范数，2-范数和无穷范数。

向量的1-范数：各个元素的绝对值之和；![\{\\left\\| X \\right\\|\_1\}][left_ X _right_1]=2+3+5+7=17；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfs1=norm(*X*,1);

向量的2-范数：每个元素的平方和再开平方根；![\{\\left\\| X \\right\\|\_2\} = \{\\left( \{\{\\rm\{2\}\} \\times \{\\rm\{2\}\} + \{\\rm\{3\}\} \\times \{\\rm\{3\}\} + \{\\rm\{5\}\} \\times \{\\rm\{5\}\} + \{\\rm\{7\}\} \\times \{\\rm\{7\}\}\} \\right)^\{\\frac\{1\}\{2\}\}\} = 9.3274][left_ X _right_2_ _ _left_ _rm_2_ _times _rm_2_ _ _rm_3_ _times _rm_3_ _ _rm_5_ _times _rm_5_ _ _rm_7_ _times _rm_7_ _right_frac_1_2_ _ 9.3274]；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfs2=norm(X,2);

向量的无穷范数：

（1）正无穷范数：向量的所有元素的绝对值中**最大**的；即*X*的正无穷范数为：7；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfsz=norm(*X*,inf);

（2）负无穷范数：向量的所有元素的绝对值中**最小**的；即*X*的负无穷范数为：2；

Matlab代码：*X*=\[2, 3, -5, -7\]; XLfsf=norm(*X*,-inf);

## **2、矩阵的范数** ##

设：向量![X \\in \{R^n\}][X _in _R_n]，矩阵![A \\in \{R^\{n \\times n\}\}][A _in _R_n _times n]，例如矩阵*A*为：

*A*=\[2, 3, -5, -7;

4, 6, 8, -4;

6, -11, -3, 16\];

**（1）矩阵的1-范数（列模）：**![\{\\left\\| A \\right\\|\_1\} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_1\}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_1\}\}\} = \\mathop \{\\max \}\\limits\_\{1 \\le j \\le n\} \\sum\\limits\_\{i = 1\}^n \{\\left| \{\{a\_\{ij\}\}\} \\right|\}][left_ A _right_1_ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_1_left_ X _right_1_ _ _mathop _max _limits_1 _le j _le n_ _sum_limits_i _ 1_n _left_ _a_ij_ _right]；矩阵的每一列上的元素绝对值先求和，再从中取个最大的，（列和最大）；即矩阵*A*的1-范数为：27

Matlab代码：fs1=norm(A,1);

**（2）矩阵的2-范数（谱模）：**![\{\\left\\| A \\right\\|\_2\} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_2\}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_2\}\}\} = \\sqrt \{\{\\lambda \_\{\\max \}\}(\{A^T\}A)\} = \\sqrt \{\\mathop \{\\max \}\\limits\_\{1 \\le i \\le n\} \\left| \{\{\\lambda \_i\}\} \\right|\}][left_ A _right_2_ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_2_left_ X _right_2_ _ _sqrt _lambda _max _A_T_A_ _ _sqrt _mathop _max _limits_1 _le i _le n_ _left_ _lambda _i_ _right]，其中 ![\{\\lambda \_i\}][lambda _i]为![\{A^T\}A][A_T_A]的特征值；矩阵![70][]的最大特征值开平方根。

Matlab代码：fs2=norm(A,2);

**（3）矩阵的无穷范数（行模）：**![\{\\left\\| A \\right\\|\_\\infty \} = \\mathop \{\\max \}\\limits\_\{X \\ne 0\} \\frac\{\{\{\{\\left\\| \{AX\} \\right\\|\}\_\\infty \}\}\}\{\{\{\{\\left\\| X \\right\\|\}\_\\infty \}\}\} = \\mathop \{\\max \}\\limits\_\{1 \\le \{\\rm\{i\}\} \\le n\} \\sum\\limits\_\{j = 1\}^n \{\\left| \{\{a\_\{ij\}\}\} \\right|\}][left_ A _right_infty _ _ _mathop _max _limits_X _ne 0_ _frac_left_ _AX_ _right_infty _left_ X _right_infty _ _ _mathop _max _limits_1 _le _rm_i_ _le n_ _sum_limits_j _ 1_n _left_ _a_ij_ _right]；矩阵的每一行上的元素绝对值先求和，再从中取个最大的，（行和最大）

Matlab代码：fswq=norm(A,inf);

下面要介绍关于机器学习中稀疏表示等一些地方用到的范数，一般有核范数，L0范数，L1范数（有时很多人也叫1范数，这就让初学者很容易混淆），L21范数（有时也叫2范数），F范数等，这些范数都是为了解决实际问题中的困难而提出的新的范数定义，不同于前面矩阵的范数。

关于核范数，L0范数，L1范数等解释见博客：

[http://www.cnblogs.com/MengYan-LongYou/p/4050862.html][http_www.cnblogs.com_MengYan-LongYou_p_4050862.html]

[https://blog.csdn.net/u013066730/article/details/51145889][https_blog.csdn.net_u013066730_article_details_51145889]

[http://blog.sina.com.cn/s/blog\_7103b28a0102w73g.html][http_blog.sina.com.cn_s_blog_7103b28a0102w73g.html]

**（4）矩阵的核范数：**矩阵的奇异值（将矩阵svd分解）之和，这个范数可以用来低秩表示（因为最小化核范数，相当于最小化矩阵的秩——低秩）；

Matlab代码：JZhfs=sum(svd(A));

**（5）矩阵的L0范数：**矩阵的非0元素的个数，通常用它来表示稀疏，L0范数越小0元素越多，也就越稀疏。

**（6）矩阵的L1范数：**矩阵中的每个元素绝对值之和，它是L0范数的最优凸近似，因此它也可以近似表示稀疏；

Matlab代码：JZL1fs=sum(sum(abs(A)));

**（7）矩阵的F范数：**矩阵的各个元素平方之和再开平方根，它通常也叫做矩阵的L2范数，它的有点在它是一个凸函数，可以求导求解，易于计算；

Matlab代码：JZFfs=norm(A,'fro');

**（8）矩阵的L21范数：**矩阵先以每一列为单位，求每一列的F范数（也可认为是向量的2范数），然后再将得到的结果求L1范数（也可认为是向量的1范数），很容易看出它是介于L1和L2之间的一种范数

Matlab代码：JZL21fs=norm(A(:,1),2) + norm(A(:,2),2) + norm(A(:,3),2)++ norm(A(:,4),2);

### **Matlab****代码** ###

clear all;clc;
    
    %% 求向量的范数
    X=[2, 3, -5, -7];   %初始化向量X
    XLfs1=norm(X,1);    %向量的1-范数
    XLfs2=norm(X,2);    %向量的2-范数
    XLfsz=norm(X,inf);  %向量的正无穷范数
    XLfsf=norm(X,-inf); %向量的负无穷范数
    
    %% 求矩阵的范数
    A=[2, 3, -5, -7;
       4, 6,  8, -4;
       6, -11, -3, 16];     %初始化矩阵A
    
    JZfs1=norm(A,1);        %矩阵的1-范数
    JZfs2=norm(A,2);        %矩阵的2-范数
    JZfswq=norm(A,inf);     %矩阵的无穷范数
    JZhfs=sum(svd(A));      %矩阵的核范数
    JZL1fs=sum(sum(abs(A)));% 矩阵的L1范数
    JZFfs=norm(A,'fro');    %矩阵的F范数
    JZL21fs=norm(A(:,1),2) + norm(A(:,2),2) + norm(A(:,3),2)++ norm(A(:,4),2);% 矩阵的L21范数

### 参考资料 ###

\[1\] [https://blog.csdn.net/Michael\_\_Corleone/article/details/75213123][https_blog.csdn.net_Michael_Corleone_article_details_75213123]

\[2\] [https://wenku.baidu.com/view/dc9e6e3753d380eb6294dd88d0d233d4b04e3f48.html][https_wenku.baidu.com_view_dc9e6e3753d380eb6294dd88d0d233d4b04e3f48.html]

\[3\] [http://www.cnblogs.com/MengYan-LongYou/p/4050862.html][http_www.cnblogs.com_MengYan-LongYou_p_4050862.html]

\[4\] [https://blog.csdn.net/u013066730/article/details/51145889][https_blog.csdn.net_u013066730_article_details_51145889]

\[5\] [http://blog.sina.com.cn/s/blog\_7103b28a0102w73g.html][http_blog.sina.com.cn_s_blog_7103b28a0102w73g.html]

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[http_www.cnblogs.com_MengYan-LongYou_p_4050862.html]: http://www.cnblogs.com/MengYan-LongYou/p/4050862.html
[https_blog.csdn.net_u013066730_article_details_51145889]: https://blog.csdn.net/u013066730/article/details/51145889
[http_blog.sina.com.cn_s_blog_7103b28a0102w73g.html]: http://blog.sina.com.cn/s/blog_7103b28a0102w73g.html
[https_blog.csdn.net_Michael_Corleone_article_details_75213123]: https://blog.csdn.net/Michael__Corleone/article/details/75213123
[https_wenku.baidu.com_view_dc9e6e3753d380eb6294dd88d0d233d4b04e3f48.html]: https://wenku.baidu.com/view/dc9e6e3753d380eb6294dd88d0d233d4b04e3f48.html
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