余弦相似度常用在文本分类、图片分类等应用中，来计算两个文本或两个图像之间的相似度。
如下图，向量
a
=
[
x
1
,
y
1
]
,
b
=
[
x
2
,
y
2
]
\boldsymbol a=[x_1,y_1],\boldsymbol b =[x_2,y_2]
a
=
[
x
1
,
y
1
]
,
b
=
[
x
2
,
y
2
]
s
i
m
(
a
,
b
)
=
c
o
s
θ
=
a
b
∣
a
∣
∣
b
∣
=
x
1
x
2
+
y
1
y
2
x
1
2
+
y
1
2
x
2
2
+
y
2
2
sim(a,b) = cos \theta = \frac {ab}{\mid a \mid \mid b \mid} = \frac {x_1x_2+y_1y_2}{\sqrt{x_1^2+y_1^2}\sqrt{x_2^2+y_2^2}}
s
i
m
(
a
,
b
)
=
c
o
s
θ
=
∣
a
∣
∣
b
∣
a
b
=
x
1
2
+
y
1
2
x
2
2
+
y
2
2
x
1
x
2
+
y
1
y
2
对于
n
n
n
维向量
A
=
[
a
1
,
a
2
,
.
.
.
a
n
]
,
B
=
[
b
1
,
b
2
,
.
.
.
b
n
]
A=[a_1,a_2,…a_n],B=[b_1,b_2,…b_n]
A
=
[
a
1
,
a
2
,
.
.
.
a
n
]
,
B
=
[
b
1
,
b
2
,
.
.
.
b
n
]
，
s
i
m
(
A
,
B
)
=
A
B
∣
A
∣
∣
B
∣
=
∑
i
=
1
n
A
i
B
i
∑
i
=
1
n
A
i
2
∑
i
=
1
n
A
i
2
sim(A,B) = \frac {AB}{\mid A \mid \mid B \mid}= \frac {\sum_{i=1}^{n}{A_iB_i}}{\sqrt{\sum_{i=1}^{n}{A_i^2}}\sqrt{\sum_{i=1}^{n}{A_i^2}}}
s
i
m
(
A
,
B
)
=
∣
A
∣
∣
B
∣
A
B
=
∑
i
=
1
n
A
i
2
∑
i
=
1
n
A
i
2
∑
i
=
1
n
A
i
B
i
余弦相似度的取值范围在-1到1之间。余弦值越接近1，也就是两个向量越相似，完全相同时数值为1；相反反向时为-1；正交或不相关是为0。
求余弦相似度需要用到
np.linalg.norm
操作，来求向量的范式，默认是L2范式，等同于求向量的欧式距离。
import numpy as np
t1 = np.array([-0.4,0.8,0.5,-0.2,0.3])
t2 = np.array([-0.5,0.4,-0.2,0.7,-0.1])
def cos_sim(a, b):
a_norm = np.linalg.norm(a)
b_norm = np.linalg.norm(b)
cos = np.dot(a,b)/(a_norm * b_norm)
return cos
print(cos_sim(t1,t2))
输出：0.23612240736068565