einsum爱因斯坦求和约定

作者: 612twilight | 来源:发表于2020-06-06 15:41 被阅读0次

einsum爱因斯坦求和约定
Einsum规则
爱因斯坦求和
numpy.einsum()
torch.einsum
PyTorch 笔记
HTTP Content-Type深入实践
爱因斯坦的感情生活
numpy.einsum-学习笔记
宗教与科学

爱因斯坦求和约定（Einstein Notation）
在数学中，爱因斯坦求和约定是一种标记法，也称为Einstein Summation Convention，在处理关于坐标的方程式时十分有效。简单来说，爱因斯坦求和就是简化掉求和式中的求和符号 ,这样就会使公式更加简洁

转置
${B_{ji}} = {A_{ij}}$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->ji', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: [[0 3 6]
 [1 4 7]
 [2 5 8]]

全部元素求和
$sum{\rm{ = }}\sum\limits_i {\sum\limits_j {{A_{ij}}} }$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: 36

某一维度求和
$sum{\rm{ = }}\sum\limits_i {{A_{ij}}}$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->i', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: [ 3 12 21]

矩阵对应维度相乘(广播形式)
${C_{ij}} = {A_{ij}}{B_j}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 4).reshape(4)
print(b)
c = np.einsum('ij,j->ij', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [0 1 2 3]
c: [[ 0  1  4  9]
 [ 0  5 12 21]
 [ 0  9 20 33]]

矩阵对应维度相乘(求和形式)
${C_i} = \sum\limits_j {{A_{ij}}{B_j}}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 4).reshape(4)
print(b)
c = np.einsum('ij,j->i', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [0 1 2 3]
c: [14 38 62]

矩阵点积
$C = \sum\limits_i {\sum\limits_j {{A_{ij}}{B_{ij}}} }$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 12).reshape(3, 4)
print(b)
c = np.einsum('ij,ij->', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
c: 506

矩阵外积(相乘)
${C_{ij}} = \sum\limits_k {{A_{ik}}{B_{kj}}}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 12).reshape(4, 3)
print(b)
c = np.einsum('ik,kj->ij', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [[ 0  1  2]
 [ 3  4  5]
 [ 6  7  8]
 [ 9 10 11]]
c: [[ 42  48  54]
 [114 136 158]
 [186 224 262]]