在python，numpy中优化矩阵运算

Question

这是一个优化问题。 给定矩阵E，H，Q，F和方法my_func_basic（参见代码块）中的逻辑，填充矩阵V。是否有任何可能的方式（例如通过矢量化）来加快计算速度？ 谢谢。

import timeit
import numpy as np
n = 20
m = 90
# E: m x n
E = np.random.randn(m,n)
# H, Q: m x m
H = np.random.randn(m,m)
Q = np.random.randn(m,m)
# F: n x n
F = np.random.randn(n,n)
# V: m x m
V = np.zeros(shape=(m,m))

def my_func_basic():
    for x in range(n):
        for y in range(n):
            if x == y:
                V[x][y] = np.nan
                continue   
            h = H[x][y]
            e = np.array([E[x,:]+h*E[y,:]])
            v1 = np.dot(np.dot(e,F),np.transpose(e))[0][0]
            v2 = Q[x][x]+h**2*Q[y][y]
            V[x][y] = v1/np.sqrt(v2)

print(timeit.timeit(my_func_basic,number=1000),'(sec), too slow...')

Answer 1

这将是使用vectorized方法解决问题的一种方法-

import numpy as np

def vectorized_approach(V,H,E,F,Q,n):

    # Create a copy of V to store output values into it
    V_vectorized = V.copy()

    # Calculate v1 in a vectorized fashion
    E1 = (E[None,:n,:]*H[:n,:n,None] + E[:n,None,:]).reshape(-1,n)
    E2 = np.dot(E1,F)
    v1_vectorized = np.einsum('ij,ji->i',E2,E1.T).reshape(n,n)
    np.fill_diagonal(v1_vectorized, np.nan)

    # Calculate v2 in a vectorized fashion
    Q_diag = np.diag(Q[:n,:n])
    v2_vectorized = Q_diag[:,None] + H[:n,:n]**2*Q_diag[None,:]

    # Finally, get vectorized version of output V
    V_vectorized[:n,:n] = v1_vectorized/np.sqrt(v2_vectorized)
    return V_vectorized 

测试：

1）设置输入-

In [314]: n = 20
     ...: m = 90
     ...: # E: m x n
     ...: E = np.random.randn(m,n)
     ...: # H, Q: m x m
     ...: H = np.random.randn(m,m)
     ...: Q = np.random.randn(m,m)
     ...: # F: n x n
     ...: F = np.random.randn(n,n)
     ...: # V: m x m
     ...: V = np.zeros(shape=(m,m))
     ...: 

2）验证结果-

In [327]: out_basic_approach = my_func_basic(V,H,E,F,Q,n)
     ...: out_vectorized_approach = vectorized_approach(V,H,E,F,Q,n)
     ...: 
     ...: mask1 = ~np.isnan(out_basic_approach)
     ...: mask2 = ~np.isnan(out_vectorized_approach)
     ...: 

In [328]: np.allclose(mask1,mask2)
Out[328]: True

In [329]: np.allclose(out_basic_approach[mask1],out_vectorized_approach[mask1])
Out[329]: True

3）运行时测试-

In [330]: %timeit my_func_basic(V,H,E,F,Q,n)
100 loops, best of 3: 12.2 ms per loop

In [331]: %timeit vectorized_approach(V,H,E,F,Q,n)
1000 loops, best of 3: 222 µs per loop