Job address
What you need to remember:
-np.exp(x) applies to any np.array x and applies the exponential function to each coordinate
-sigmoid function and its gradient
-image2vector is usually used for deep learning
-np.reshape is widely used. Keeping the matrix/vector size constant helps us eliminate many errors.
-numpy has efficient built-in functions
-broadcasting is very useful
The realization of vectorization is more concise and efficient. For larger vectors/matrices, the difference in running time becomes greater. (Mainly use some built-in functions of numpy)
In deep learning, usually very large data sets need to be processed. Therefore, the non-computed optimal function may become a huge bottleneck in the algorithm and may cause the model to run for a period of time. In order to ensure efficient calculation of the code, we will use vectorization. For example, try to distinguish the difference between dot/outer/element product.
import time
x1 = [9, 2, 5, 0, 0, 7, 5, 0, 0, 0, 9, 2, 5, 0, 0]
x2 = [9, 2, 2, 9, 0, 9, 2, 5, 0, 0, 9, 2, 5, 0, 0]
### CLASSIC DOT PRODUCT OF VECTORS IMPLEMENTATION ###
tic = time.process_time()
dot = 0
for i in range(len(x1)):
dot+= x1[i]*x2[i]
toc = time.process_time()
print ("dot = " + str(dot) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### CLASSIC OUTER PRODUCT IMPLEMENTATION ###
tic = time.process_time()
outer = np.zeros((len(x1),len(x2))) # we create a len(x1)*len(x2) matrix with only zeros
for i in range(len(x1)):
for j in range(len(x2)):
outer[i,j] = x1[i]*x2[j]
toc = time.process_time()
print ("outer = " + str(outer) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### CLASSIC ELEMENTWISE IMPLEMENTATION ###
tic = time.process_time()
mul = np.zeros(len(x1))
for i in range(len(x1)):
mul[i] = x1[i]*x2[i]
toc = time.process_time()
print ("elementwise multiplication = " + str(mul) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### CLASSIC GENERAL DOT PRODUCT IMPLEMENTATION ###
W = np.random.rand(3,len(x1)) # Random 3*len(x1) numpy array
tic = time.process_time()
gdot = np.zeros(W.shape[0])
for i in range(W.shape[0]):
for j in range(len(x1)):
gdot[i] += W[i,j]*x1[j]
toc = time.process_time()
print ("gdot = " + str(gdot) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
Output:
dot = 278
----- Computation time = 0.08447300000002933ms
outer = [[81. 18. 18. 81. 0. 81. 18. 45. 0. 0. 81. 18. 45. 0. 0.]
[18. 4. 4. 18. 0. 18. 4. 10. 0. 0. 18. 4. 10. 0. 0.]
[45. 10. 10. 45. 0. 45. 10. 25. 0. 0. 45. 10. 25. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[63. 14. 14. 63. 0. 63. 14. 35. 0. 0. 63. 14. 35. 0. 0.]
[45. 10. 10. 45. 0. 45. 10. 25. 0. 0. 45. 10. 25. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[81. 18. 18. 81. 0. 81. 18. 45. 0. 0. 81. 18. 45. 0. 0.]
[18. 4. 4. 18. 0. 18. 4. 10. 0. 0. 18. 4. 10. 0. 0.]
[45. 10. 10. 45. 0. 45. 10. 25. 0. 0. 45. 10. 25. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]
----- Computation time = 0.22404599999992225ms
elementwise multiplication = [81. 4. 10. 0. 0. 63. 10. 0. 0. 0. 81. 4. 25. 0. 0.]
----- Computation time = 0.10582699999994727ms
gdot = [26.19713459 12.20793127 23.40980652]
----- Computation time = 0.15482099999997168ms
x1 = [9, 2, 5, 0, 0, 7, 5, 0, 0, 0, 9, 2, 5, 0, 0]
x2 = [9, 2, 2, 9, 0, 9, 2, 5, 0, 0, 9, 2, 5, 0, 0]
### VECTORIZED DOT PRODUCT OF VECTORS ###
tic = time.process_time()
dot = np.dot(x1,x2)
toc = time.process_time()
print ("dot = " + str(dot) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### VECTORIZED OUTER PRODUCT ###
tic = time.process_time()
outer = np.outer(x1,x2)
toc = time.process_time()
print ("outer = " + str(outer) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### VECTORIZED ELEMENTWISE MULTIPLICATION ###
tic = time.process_time()
mul = np.multiply(x1,x2)
toc = time.process_time()
print ("elementwise multiplication = " + str(mul) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
### VECTORIZED GENERAL DOT PRODUCT ###
tic = time.process_time()
dot = np.dot(W,x1)
toc = time.process_time()
print ("gdot = " + str(dot) + "\n ----- Computation time = " + str(1000*(toc - tic)) + "ms")
Output:
dot = 278
----- Computation time = 0.0ms
outer = [[81 18 18 81 0 81 18 45 0 0 81 18 45 0 0]
[18 4 4 18 0 18 4 10 0 0 18 4 10 0 0]
[45 10 10 45 0 45 10 25 0 0 45 10 25 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[63 14 14 63 0 63 14 35 0 0 63 14 35 0 0]
[45 10 10 45 0 45 10 25 0 0 45 10 25 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[81 18 18 81 0 81 18 45 0 0 81 18 45 0 0]
[18 4 4 18 0 18 4 10 0 0 18 4 10 0 0]
[45 10 10 45 0 45 10 25 0 0 45 10 25 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]]
----- Computation time = 0.0ms
elementwise multiplication = [81 4 10 0 0 63 10 0 0 0 81 4 25 0 0]
----- Computation time = 0.0ms
gdot = [ 21.57937154 22.58814194 13.70092277]
----- Computation time = 0.0ms
The following is some code in the homework, you can look at it or not:
import numpy as np
def sigmoid(x):
return 1 / (1 + np.exp(-x))
def sigmoid_grad(x):
return sigmoid(x) * (1 - sigmoid(x))
def image2vector(image):
v = image.reshape(3 * 3 * 2, 1)
return v
def normalizeRows(x):
x_norm = np.linalg.norm(x, axis=1, keepdims=True)
return x / x_norm # python中有广播 所以可以直接除
def softmax(x):
# x = np.array([
# [9, 2, 5, 0, 0],
# [7, 5, 0, 0, 0]])
x_exp = np.exp(x)
x_sum = np.sum(x_exp, axis=1, keepdims=True)
s = x_exp / x_sum # python中有广播 所以可以直接除
return s