PyTorch深度学习入门 Tensor基础 1.1 Tensor 5. 线性代数运算

迈不过友情╰ 2024-03-27 15:31 89阅读 0赞

#### PyTorch深度学习入门 ####

![在这里插入图片描述][689fa44313724705803dba57df8e0922.png_pic_center]

##### Tensor基础 #####

###### 5. 线性代数运算 ######

简单介绍3种：

使用torch.dot()函数实现向量与向量的点积（也称内积：

a = torch.Tensor([1, 2, 3])
    b = torch.Tensor([2, 3, 4])
    torch.dot(a, b)

![在这里插入图片描述][5ad1fe9445a6447390b89cf7381b94dc.png_pic_center]

向量的点积定义：

假设有两个向量  a ⃗ = \[ a 1 , a 2 , ⋯   , a n \] , b ⃗ = \[ b 1 , b 2 , ⋯   , b n \] \\vec\{a\}=\\left\[a\_1, a\_2, \\cdots, a\_n\\right\], \\vec\{b\}=\\left\[b\_1, b\_2, \\cdots, b\_n\\right\] a=\[a1,a2,⋯,an\],b=\[b1,b2,⋯,bn\], 那么  a ⃗ \\vec\{a\} a 与  b ⃗ \\vec\{b\} b 的点积定义为:  
 a ⃗ ⋅ b ⃗ = a 1 b 1 + a 2 b 2 + ⋯ + a n b n = ∑ i = 1 n a i b i \\vec\{a\} \\cdot \\vec\{b\}=a\_1 b\_1+a\_2 b\_2+\\cdots+a\_n b\_n=\\sum\_\{i=1\}^n a\_i b\_i a⋅b=a1b1\+a2b2\+⋯\+anbn=i=1∑naibi

所以  a ⃗ ⋅ b ⃗ = 1 × 2 + 2 × 3 + 3 × 4 = 20 \\vec\{a\} \\cdot \\vec\{b\}=1 \\times 2+2 \\times 3+3 \\times 4=20 a⋅b=1×2\+2×3\+3×4=20

使用torch.mv()函数实现矩阵与向量的乘法：

![在这里插入图片描述][4c77a101d07640279cab38a24760edab.png_pic_center]

矩阵与向量的乘法规则如下：  
 \[ a 11 a 12 ⋯ a 1 m a 21 a 22 ⋯ a 2 m ⋮ ⋮ ⋱ ⋮ a n 1 a n 2 ⋯ a n m \] \[ b 1 b 2 ⋮ b m \] = \[ a 11 b 1 + a 12 b 2 + ⋯ + a 1 m b m a 21 b 1 + a 22 b 2 + ⋯ + a 2 m b m ⋮ a n 1 b 1 + a n 2 b 2 + ⋯ + a n m b m \] \\left\[\\begin\{array\}\{cccc\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 m\} \\\\ a\_\{21\} & a\_\{22\} & \\cdots & a\_\{2 m\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{n 1\} & a\_\{n 2\} & \\cdots & a\_\{n m\} \\end\{array\}\\right\]\\left\[\\begin\{array\}\{c\} b\_1 \\\\ b\_2 \\\\ \\vdots \\\\ b\_m \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{c\} a\_\{11\} b\_1+a\_\{12\} b\_2+\\cdots+a\_\{1 m\} b\_m \\\\ a\_\{21\} b\_1+a\_\{22\} b\_2+\\cdots+a\_\{2 m\} b\_m \\\\ \\vdots \\\\ a\_\{n 1\} b\_1+a\_\{n 2\} b\_2+\\cdots+a\_\{n m\} b\_m \\end\{array\}\\right\] ⎣⎢⎢⎢⎡a11a21⋮an1a12a22⋮an2⋯⋯⋱⋯a1ma2m⋮anm⎦⎥⎥⎥⎤⎣⎢⎢⎢⎡b1b2⋮bm⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎡a11b1\+a12b2\+⋯\+a1mbma21b1\+a22b2\+⋯\+a2mbm⋮an1b1\+an2b2\+⋯\+anmbm⎦⎥⎥⎥⎤

所以  
 \[ 1 2 3 2 3 4 3 4 5 \] \[ 1 2 3 \] = \[ 1 × 1 + 2 × 2 + 3 × 3 2 × 1 + 3 × 2 + 4 × 3 3 × 1 + 4 × 2 + 5 × 3 \] = \[ 14 20 26 \] \\left\[\\begin\{array\}\{lll\} 1 & 2 & 3 \\\\ 2 & 3 & 4 \\\\ 3 & 4 & 5 \\end\{array\}\\right\]\\left\[\\begin\{array\}\{l\} 1 \\\\ 2 \\\\ 3 \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{l\} 1 \\times 1+2 \\times 2+3 \\times 3 \\\\ 2 \\times 1+3 \\times 2+4 \\times 3 \\\\ 3 \\times 1+4 \\times 2+5 \\times 3 \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{l\} 14 \\\\ 20 \\\\ 26 \\end\{array\}\\right\] ⎣⎡123234345⎦⎤⎣⎡123⎦⎤=⎣⎡1×1\+2×2\+3×32×1\+3×2\+4×33×1\+4×2\+5×3⎦⎤=⎣⎡142026⎦⎤

使用torch.mm()函数将两个矩阵相乘：

![在这里插入图片描述][6f1729bda0ed422f90165caac96a93a5.png_pic_center]

矩阵与矩阵的乘法定义：

\[ a 11 a 12 ⋯ a 1 s ⋮ ⋮ ⋱ ⋮ a i 1 a i 2 ⋯ a i s ⋮ ⋮ ⋱ ⋮ a n 1 a n 2 ⋯ a n s \] \[ b 11 ⋯ b 1 j ⋯ b 1 m b 21 ⋯ b 2 j ⋯ b 2 m ⋮ ⋱ ⋮ ⋯ ⋮ b s 1 ⋯ b s j ⋯ b s m \] = \[ c 11 ⋯ c 1 j ⋯ c 1 m ⋮ ⋮ ⋮ ⋱ ⋮ c i 1 ⋯ c i j ⋯ c i m ⋮ ⋮ ⋮ ⋱ ⋮ c n 1 ⋯ c n j ⋯ c n m \] \\left\[\\begin\{array\}\{cccc\} a\_\{11\} & a\_\{12\} & \\cdots & a\_\{1 s\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{i 1\} & a\_\{i 2\} & \\cdots & a\_\{i s\} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_\{n 1\} & a\_\{n 2\} & \\cdots & a\_\{n s\} \\end\{array\}\\right\]\\left\[\\begin\{array\}\{ccccc\} b\_\{11\} & \\cdots & b\_\{1 j\} & \\cdots & b\_\{1 m\} \\\\ b\_\{21\} & \\cdots & b\_\{2 j\} & \\cdots & b\_\{2 m\} \\\\ \\vdots & \\ddots & \\vdots & \\cdots & \\vdots \\\\ b\_\{s 1\} & \\cdots & b\_\{s j\} & \\cdots & b\_\{s m\} \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{ccccc\} c\_\{11\} & \\cdots & c\_\{1 j\} & \\cdots & c\_\{1 m\} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ c\_\{i 1\} & \\cdots & c\_\{i j\} & \\cdots & c\_\{i m\} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ c\_\{n 1\} & \\cdots & c\_\{n j\} & \\cdots & c\_\{n m\} \\end\{array\}\\right\] ⎣⎢⎢⎢⎢⎢⎢⎡a11⋮ai1⋮an1a12⋮ai2⋮an2⋯⋱⋯⋱⋯a1s⋮ais⋮ans⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎡b11b21⋮bs1⋯⋯⋱⋯b1jb2j⋮bsj⋯⋯⋯⋯b1mb2m⋮bsm⎦⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡c11⋮ci1⋮cn1⋯⋮⋯⋮⋯c1j⋮cij⋮cnj⋯⋱⋯⋱⋯c1m⋮cim⋮cnm⎦⎥⎥⎥⎥⎥⎥⎤

其中  c i j = a i 1 b 1 j + a i 2 b 2 j + ⋯ + a i s b s j c\_\{i j\}=a\_\{i 1\} b\_\{1 j\}+a\_\{i 2\} b\_\{2 j\}+\\cdots+a\_\{i s\} b\_\{s j\} cij=ai1b1j\+ai2b2j\+⋯\+aisbsj 。

所以：

\[ 1 2 3 2 3 4 3 4 5 \] \[ 2 3 4 3 4 5 4 5 6 \] = \[ 20 26 32 29 38 47 38 50 62 \] \\left\[\\begin\{array\}\{lll\} 1 & 2 & 3 \\\\ 2 & 3 & 4 \\\\ 3 & 4 & 5 \\end\{array\}\\right\]\\left\[\\begin\{array\}\{lll\} 2 & 3 & 4 \\\\ 3 & 4 & 5 \\\\ 4 & 5 & 6 \\end\{array\}\\right\]=\\left\[\\begin\{array\}\{lll\} 20 & 26 & 32 \\\\ 29 & 38 & 47 \\\\ 38 & 50 & 62 \\end\{array\}\\right\] ⎣⎡123234345⎦⎤⎣⎡234345456⎦⎤=⎣⎡202938263850324762⎦⎤

常见的线性代数操作：

<table> 
 <thead> 
  <tr> 
   <th align="center">方法</th> 
   <th align="center">说明</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td align="center">dot()</td> 
   <td align="center">两个向量点积</td> 
  </tr> 
  <tr> 
   <td align="center">mv()</td> 
   <td align="center">矩阵与向量相乘</td> 
  </tr> 
  <tr> 
   <td align="center">mm()</td> 
   <td align="center">两个矩阵相乘</td> 
  </tr> 
  <tr> 
   <td align="center">addmm()</td> 
   <td align="center">将两个矩阵进行矩阵乘法操作的结果与另一矩阵相加</td> 
  </tr> 
  <tr> 
   <td align="center">addmv()</td> 
   <td align="center">将矩阵和向量相乘的结果与另一向量相加</td> 
  </tr> 
  <tr> 
   <td align="center">addr()</td> 
   <td align="center">将两个向量进行张量积（外积）操作的结果与另一个矩阵相加</td> 
  </tr> 
  <tr> 
   <td align="center">bmm()</td> 
   <td align="center">两个batch内的矩阵进行批矩阵乘法</td> 
  </tr> 
  <tr> 
   <td align="center">eig()</td> 
   <td align="center">计算方阵的特征值和特征向量</td> 
  </tr> 
  <tr> 
   <td align="center">ger()</td> 
   <td align="center">两个向量的张量积</td> 
  </tr> 
  <tr> 
   <td align="center">inverse()</td> 
   <td align="center">对方阵求逆</td> 
  </tr> 
  <tr> 
   <td align="center">addbmm()</td> 
   <td align="center">将两个batch内的矩阵进行批矩阵乘法操作并累加，其结果与另一矩阵相加</td> 
  </tr> 
  <tr> 
   <td align="center">naddbmm()</td> 
   <td align="center">将两个batch内的矩阵进行批矩阵乘法操作，其结果与另一batch内的矩阵相加</td> 
  </tr> 
 </tbody> 
</table>

[689fa44313724705803dba57df8e0922.png_pic_center]: https://image.dandelioncloud.cn/pgy_files/images/2024/03/27/ef52355462c644a7bb674bc95d768d28.png
[5ad1fe9445a6447390b89cf7381b94dc.png_pic_center]: https://image.dandelioncloud.cn/pgy_files/images/2024/03/27/182fb8cffef543809ba59569308224ae.png
[4c77a101d07640279cab38a24760edab.png_pic_center]: https://image.dandelioncloud.cn/pgy_files/images/2024/03/27/011ea692886b40dd8b67f04474966b00.png
[6f1729bda0ed422f90165caac96a93a5.png_pic_center]: https://image.dandelioncloud.cn/pgy_files/images/2024/03/27/8336999d09b84339a8d09b8881efc02b.png