ⴰⴳⵎⵓⴹ ⵏ Einstein

ⵜⴰⵙⴻⴽⴽⵉⵔⵜ ⵏ Einstein Sumation ⴷ ⴰⴱⵔⵉⴷ ⵉⴳⴻⵊⴷⴰⵏⴻⵏ ⵢⴻⵔⵏⴰ ⴷ ⴰⵊⴻⵀⴷⴻⵏ ⵉ ⵓⵙⴻⵎⵔⴻⵙ ⵏ ⵜⵎⴻⵥⵔⵉⵡⵉⵏ ⵏ tensor, ⵢⴻⵜⵜⵡⴰⵙⴻⵇⴷⴰⵛ ⵙ ⵡⴰⵟⴰⵙ ⴷⴻⴳ ⵜⴼⵉⵣⵉⴽⵜ ⴷ ⵓⵍⵎⴰⴷ ⵏ ⵜⵎⴰⵛⵉⵏⵉⵏ. ⵢⴻⵜⵜⴰⴵⴵⴰⴰⵖ ⴰⴷ ⵏⴰⵔⵓ ⵍⴻⵃⵙⴰⴱⴰⵜ ⵉⵅⵓⵚⵚⴻⵏ ⵖⴻⴼ ⵢⵉⵜⴻⵏⵙⵓⵔⴻⵏ ⵙ ⵜⴰⵍⵖⴰ ⵏ ⵜⵎⴻⵥⵥⵓⵖⵜ. ⴰⴷ ⴷⵏⴻⴱⴷⴻⵔ ⵍⵙⴰⵙ ⵖⴻⴼ ⵓⵙⴷⵓⴽⴽⴻⵍ ⵏ Einstein, ⴰⵎⴻⴽ ⴰⵔⴰ ⵜⵏⴻⵙⵙⴻⵅⴷⴻⵎ ⴷⴻⴳ Python ⵙ Numpy ⴷ Tensorflow, ⴰⴷ ⴷⵏⴻⴼⴽ ⵉⵎⴻⴷⵢⴰⵜⴻⵏ ⴰⵔⴰ ⴷⵢⴻⵙⵙⴽⴻⵏ ⴰⵙⴻⵇⴷⴻⵛⵉⵙ.

ⵉⵍⵓⴳⴰⵏ ⵏ ⵓⵙⴷⵓⴽⴽⴻⵍ ⵏ Einstein

ⵜⴰⵙⴻⴽⴽⵉⵔⵜ ⵏ ⵓⵙⴷⵓⴽⴽⴻⵍ ⵏ Einstein (Einsum) ⵜⴻⵜⵜⵡⴰⵙⴱⴻⴷⴷ ⵖⴻⴼ ⵜⵎⵓⵖⵍⵉ ⵏ ⵓⵙⴷⵓⴽⴽⴻⵍ ⵖⴻⴼ ⵢⵉⴹⵔⵉⵙⴻⵏ ⵢⴻⵜⵜⵡⴰⵄⴻⵔⴹⴻⵏ ⴷⴻⴳ ⵜⵎⴻⵙⵍⴰⵢⵉⵏ ⵏ tensor. ⵢⴻⵜⵜⵡⴰⴱⵏⴰ ⵖⴻⴼ ⵙⵉⵏ ⵏ ⵢⵉⵍⵓⴳⴰⵏⴰ :

1. ⴰⵙⴻⵎⵔⴻⵙ ⵖⴻⴼ ⵢⵉⴹⵔⵉⵙⴻⵏ ⵢⴻⵜⵜⵡⴰⵄⴻⵔⴹⴻⵏ: Ma ⵢⴻⵍⵍⴰ ⴰⴹⵔⵉⵙ ⵢⴻⵜⵜⴱⴰⵏⴷ ⵙⵏⴰⵜ ⵏ ⵜⵉⴽⴽⴰⵍ ⴷⴻⴳ ⵢⵉⵡⴻⵏ ⵏ ⵓⵎⵢⴰⴳ, ⵢⴻⵜⵜⵡⴰⵙⵙⵓⵎⵎⴻⵍ ⵖⴻⴼ .

2. ⵉⴹⵔⵉⵙⴻⵏ ⵉⵍⴻⵍⵍⵉⵢⴻⵏ: ⵉⴹⵔⵉⵙⴻⵏ ⵉ ⴷⵢⴻⵜⵜⴱⴰⵏⴻⵏ ⵢⵉⵡⴻⵜ ⴽⴰⵏ ⵏ ⵜⵉⴽⴽⴻⵍⵜ ⴷ ⵉⴹⵔⵉⵙⴻⵏ ⵉⵍⴻⵍⵍⵉⵢⴻⵏ ⵢⴻⵔⵏⴰ ⵜⵜⴳⴻⵏⵙⵉⵙⴻⵏⴷ ⵉⵅⴻⴼ ⵏ ⵜⴻⵏⵙⵓⵔ ⵏ ⵓⵙⵍⵓⴳⴻⵏ .

ⴰⴷ ⴷⵏⴻⵙⵎⴻⴽⵜⵉ ⴰⵢⴰ ⵙ ⵓⵎⴻⴷⵢⴰ ⵏ ⵓⵙⵏⴻⵔⵏⵉ ⵏ ⵙⵏⴰⵜ ⵏ ⵜⵎⴰⵜⵔⵉⵢⵉⵏ A ⴷ B : ⵜⴰⵎⴰⵜⵔⵉⵙⵜ C ⵉ ⴷⵢⴻⵜⵜⵡⴰⵙⵏⵓⵍⴼⴰⵏ ⵜⴻⵜⵜⵡⴰⵙⵙⴻⵏ ⵙ .

$C_{ik} = \sum\limits_{j}^{}A_{ij}B_{jk}$

ⴷⴻⴳ Pitun, ⴰⵎⴰ ⴷ ⵜⵉⵎⴽⴻⵔⴹⵉⵡⵉⵏ ⵏ Numpy ⴰⵎⴰ ⴷ ⵜⵉⵎⴽⴻⵔⴹⵉⵡⵉⵏ ⵏ Tensorflow ⵜⵜⴰⴽⴻⵏⵜⴷ ⵜⴰⵖⵓⵍⵜ ⵏ ⴻⵉⵏⵙⵓⵎ.

ⵏⵓⵎⴱⵉ

import numpy as np

# Define two matrices A and B
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Perform matrix multiplication using einsum
C = np.einsum('ij,jk->ik', A, B)

print(C)
# [[19 22]
#  [43 50]]

ⴷⴻⴳ ⵓⵎⴻⴷⵢⴰ ⵏ ⵜⴼⴻⵍⵡⵉⵜ, ⵉⵊ,ⵊⴽ->ⵉⴽ ⴷ ⵜⴰⵖⴻⵛⵜ ⵏ ⴻⵉⵏⵙⵓⵎ:

ⵉⵊ ⵢⴻⵜⵜⴳⴻⵏⵙⵉⵙⴷ ⵜⵉⵡⵜⵉⵍⵉⵏ ⵏ ⵎⴰⵜⵔⵉⵅ A

ⵊⴽ ⵢⴻⵜⵜⴳⴻⵏⵙⵉⵙⴷ ⵜⵉⵡⵜⵉⵍⵉⵏ ⵏ ⵎⴰⵜⵔⵉⵅ B

->ⵉⴽ ⵢⴻⵙⵙⴽⴰⵏⴰⵢⴷ ⵉⵏⴷⴻⵅ ⵏ ⵜⵎⴰⵜⵔⵉⵜ ⵏ ⵓⵙⵍⵓⴳⴻⵏ C

ⵜⴰⵎⵀⴻⵍⵜ ⵜⴻⵜⵜⴽⴻⵎⵎⵉⵍ ⵖⴻⴼ ⵓⵎⴹⵉⵇ ⵊ

ⴰⵎ ⵡⴰⴽⴽⴻⵏ ⴰⵔⴰ ⵢⵉⵍⵉ ⵓⴽⴰⵔⴰⵎⵏⵏⵉⴹⴻⵏ ⴷⴻⴳ Tensorflow

import tensorflow as tf

# Define two matrices A and B
A = tf.constant([[1, 2], [3, 4]], dtype=tf.float32)
B = tf.constant([[5, 6], [7, 8]], dtype=tf.float32)

# Perform matrix multiplication using einsum
C = tf.einsum('ij,jk->ik', A, B)

print(C)
# tf.Tensor(
# [[19. 22.]
#  [43. 50.]], shape=(2, 2), dtype=float32)

ⵓⴳⴰⵔ ⵏ ⵉⵎⴻⴷⵢⴰⵜⴻⵏ

ⴰⴼⴰⵔⵉⵙ ⵏ ⴷⴰⵅⴻⵍ ⵏ ⵢⵉⵙⴻⴽⴽⵉⵍⴻⵏ

ⴰⴼⴰⵔⵉⵙ ⵏ ⴷⴰⵅⴻⵍ (ⴰⴼⴰⵔⵉⵙ ⵏ ⵜⵏⴻⵇⴹⵉⵏ) ⵏ ⵙⵉⵏ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏ ⴰ ⴷ ⴱ ⵢⴻⵜⵜⵡⴰⵙⴱⴻⴳⴳⴻⵏⴷ ⵙ

$c = \sum\limits_{i}^{}a_{i}b_{i}$

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

c = np.einsum('i,i->', a, b)

print(c)  # Output: 32

ⴰⴼⴰⵔⵉⵙ ⵏ ⴱⴻⵕⵕⴰ ⵏ ⵉⵎⵙⴻⴹⵔⵓⵢⴻⵏ

ⴰⴼⴰⵔⵉⵙ ⵏ ⴱⴻⵔⵔⴰ ⵏ ⵙⵉⵏ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏ ⴰ ⴷ ⴱ ⵢⴻⵜⵜⵓⵏⴻⴼⴽ ⵙ:

$C_{ij} = a_{i}b_{j}$

C = np.einsum('i,j->ij', a, b)

print(C)
# Output
# [[4 5 6]
#  [8 10 12]
#  [12 15 18]]

ⴰⵙⵉⵡⴻⴹ ⵏ Matrix

ⴰⵙⴻⴽⵛⴻⵎ ⵏ ⵜⵎⴰⵜⵔⵉⵜ A ⵢⴻⵣⵎⴻⵔ ⴰⴷ ⴷⵢⴻⵜⵜⵡⴰⴼⴽ ⵙ ⵓⴱⴻⴷⴷⴻⵍ ⵏ ⵢⵉⴹⵔⵉⵙⴻⵏⵉⵙ

A_transpose = np.einsum('ij->ji', A)

print(A_transpose)
# Output
# [[1. 3.]
#  [2. 4.]]

ⴰⴹⴰⵔ ⵏ Matrix

ⴰⴹⴰⵔ ⵏ ⵎⴰⵜⵔⵉⵅ A ⴷ ⴰⴳⵎⵓⴹ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏⵉⵙ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ:

$Tr(A) = \sum\limits_{i}^{}A_{ii}A_{ii}$

trace = np.einsum('ii->', A)

print(trace)
# Output: 5.0

ⴰⵙⵏⴻⵔⵏⵉ ⵏ Matrix ⵏ ⵜⴼⴻⵔⴽⵉⵜ

ⴻⵉⵏⵙⵓⵎ ⴷ ⴰⵢⴻⵏ ⵉ ⴷⵢⴻⵜⵜⴰⵡⵉⵏ ⵏⵏⴼⴻⵄ ⵍⴰⴷⵖⴰ ⵉ ⵜⵎⴻⵥⵔⵉⵡⵉⵏ ⵏ ⵜⴼⴻⵔⴽⵉⵜ. ⴰⴷ ⴷⵏⵉⵏⵉ ⴱⴻⵍⵍⵉ ⵏⴻⵙⵄⴰ ⵢⵉⵡⴻⵜ ⵏ ⵜⴼⴻⵔⴽⵉⵜ ⵏ ⵜⵎⴰⵜⵔⵉⵢⵉⵏ A ⴷ B, ⵢⴻⵔⵏⴰ ⵏⴻⴱⵖⴰ ⴰⴷ ⵏⴻⵙⵙⴻⵎⵔⴻⵙ ⵜⵉⵎⴰⵜⵔⵉⵢⵉⵏ ⵉ ⴷⵢⴻⵜⵜⵡⴰⵙⵇⴻⴷⵛⴻⵏ ⴷⴻⴳ ⵜⴼⴻⵔⴽⵉⵜⵏⵏⵉ:

A = np.random.rand(3, 2, 2)
B = np.random.rand(3, 2, 2)

# Perform batch matrix multiplication
C = np.einsum('bij,bjk->bik', A, B)

print(C)

ⴷⴰⴳⵉ, ⴱ ⵢⴻⵜⵜⴳⴻⵏⵙⵉⵙⴷ ⵍⵇⵉⴷⴰⵔ ⵏ ⵜⴼⴻⵔⴽⵉⵜ.

ⵍⴼⴰⵢⴷⴰⵜ ⵏ ⵓⵙⴻⴽⵍⴻⵙ ⵏ Einsum

1. ⵜⴰⴳⵣⴻⵎⵜ: Tagzemt ⵏ Einsum ⴷ ⵜⴰⵎⴻⵥⵢⴰⵏⵜ, ⵢⴻⵔⵏⴰ ⵜⴻⵣⵎⴻⵔ ⴰⴷ ⴷⵜⴻⵙⵙⴽⴻⵏ ⵜⵉⵎⵀⴰⵍ ⵙ ⵜⵖⴻⵣⴼⴰⵏⵜ

2. ⴼⵍⴻⵅⵉⴱⵉⵍⵉⵜⵢ: Yezmer ⴰⴷ ⵢⴻⵃⵔⴻⵣ ⴰⵟⴰⵙ ⵏ ⵜⵖⴰⵡⵙⵉⵡⵉⵏ ⵏ tensor ⵢⴻⵎⴳⴰⵔⴰⴷⴻⵏ ⵡⴰⵔ ⵎⴰ ⵢⴻⵜⵜⴱⴻⴷⴷⵉⵍ ⵙ ⵜⵖⴰⵡⵍⴰ ⵏⴻⵖ ⵢⴻⵜⵜⴱⴻⴷⴷⵉⵍⴻⵏ ⵜⵉⴼⴻⵍⵡⵉⵢⵉⵏ .

3. ⴻⴼⴼⵉⵛⵉⴻⵏⵛⴻ: ⴰⵟⴰⵙ ⵏ ⵜⴻⵎⴽⴰⵔⴹⵉⵢⵉⵏ ⵉ ⵢⴻⵙⵙⴻⵎⵖⴰⵔⴻⵏ ⵜⵉⵎⵀⴰⵍ ⵏ ⴻⵉⵏⵙⵓⵎ ⵙ ⴷⴰⵅⴻⵍ, ⵢⴻⵣⵎⴻⵔ ⴰⴷ ⴷⵢⴰⵡⵉ ⵙ ⵜⵖⴰⵔⴰ ⵢⴻⵍⵀⴰⵏ.