python - 如何找到一个向量与矩阵的余弦相似度

Question

我有一个形状为 (149,1001) 的 TF-IDF 矩阵。想要的是计算最后一列的余弦相似度，所有列
这是我所做的

from numpy import dot
from numpy.linalg import norm
for i in range(mat.shape[1]-1):
    cos_sim = dot(mat[:,i], mat[:,-1])/(norm(mat[:,i])*norm(mat[:,-1]))
    cos_sim

但是这个循环让它变慢了。那么，有没有什么有效的方法呢？我只想用 numpy

Answer 1

杠杆2D矢量化 matrix-multiplication
这是 NumPy 对二维数据使用矩阵乘法的一个 -

p1 = mat[:,-1].dot(mat[:,:-1])
p2 = norm(mat[:,:-1],axis=0)*norm(mat[:,-1])
out1 = p1/p2

说明: p1是 dot(mat[:,i], mat[:,-1]) 循环的矢量化等价物. p2是 (norm(mat[:,i])*norm(mat[:,-1])) .
用于验证的 sample 运行 -

In [57]: np.random.seed(0)
    ...: mat = np.random.rand(149,1001)

In [58]: out = np.empty(mat.shape[1]-1)
    ...: for i in range(mat.shape[1]-1):
    ...:     out[i] = dot(mat[:,i], mat[:,-1])/(norm(mat[:,i])*norm(mat[:,-1]))

In [59]: p1 = mat[:,-1].dot(mat[:,:-1])
    ...: p2 = norm(mat[:,:-1],axis=0)*norm(mat[:,-1])
    ...: out1 = p1/p2

In [60]: np.allclose(out, out1)
Out[60]: True

时间——

In [61]: %%timeit
    ...: out = np.empty(mat.shape[1]-1)
    ...: for i in range(mat.shape[1]-1):
    ...:     out[i] = dot(mat[:,i], mat[:,-1])/(norm(mat[:,i])*norm(mat[:,-1]))
18.5 ms ± 977 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

In [62]: %%timeit   
    ...: p1 = mat[:,-1].dot(mat[:,:-1])
    ...: p2 = norm(mat[:,:-1],axis=0)*norm(mat[:,-1])
    ...: out1 = p1/p2
939 µs ± 29.2 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

# @yatu's soln
In [89]: a = mat

In [90]: %timeit cosine_similarity(a[None,:,-1] , a.T[:-1])
2.47 ms ± 461 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

进一步优化 norm与 einsum
或者，我们可以计算 p2与 np.einsum .
所以，norm(mat[:,:-1],axis=0)可以替换为:

np.sqrt(np.einsum('ij,ij->j',mat[:,:-1],mat[:,:-1]))

因此，给我们一个修改后的 p2 :

p2 = np.sqrt(np.einsum('ij,ij->j',mat[:,:-1],mat[:,:-1]))*norm(mat[:,-1])

与之前相同设置的时间 -

In [82]: %%timeit
    ...: p1 = mat[:,-1].dot(mat[:,:-1])
    ...: p2 = np.sqrt(np.einsum('ij,ij->j',mat[:,:-1],mat[:,:-1]))*norm(mat[:,-1])
    ...: out1 = p1/p2
607 µs ± 132 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

30x+ 加速超过循环!