1 餘弦相似度
餘弦相似度是一種我們較爲常用的計算向量相似度的方法。下面就是計算餘弦相似度的公式：
import torch
import torch.nn as nn
import math
class ConineSimilarity(nn.Module):
def forward(self, tensor_1,tensor_2):
normalized_tensor_1=tensor_1/tensor_1.norm(dim=-1,keepdim=True)
normalized_tensor_2=tensor_2/tensor_2.norm(dim=-1,keepdim=True)
return (normalized_tensor_1*normalized_tensor_2).sum(dim=-1)
input_1=torch.randn(3,5,requires_grad=True)
print(input_1)
input_2=torch.randn(3,5,requires_grad=True)
print(input_2)
con=ConineSimilarity()
CS=con(input_1,input_2)
CS2=torch.cosine_similarity(input_1,input_2)
print(CS)
print(CS2)
輸出：
tensor([[ 0.5973, 0.1187, -0.8476, 1.1966, 0.8405],
[ 0.4578, -0.1856, -0.7641, 0.1625, -2.2154],
[-0.2436, -1.2263, 2.2282, 0.5284, 0.4729]], requires_grad=True)
tensor([[ 0.6518, -0.3185, -0.7898, 0.0890, 0.5226],
[-1.8873, 0.7438, 0.5145, -1.0182, 2.0877],
[ 0.5931, -0.5107, -1.3426, 1.4966, -1.9130]], requires_grad=True)
tensor([ 0.7290, -0.8242, -0.3432], grad_fn=<SumBackward2>)
tensor([-0.8748, -0.5227, -0.5036], grad_fn=<DivBackward0>)
2 點乘相似度
計算公式如下：
class DotProductSimilarity(nn.Module):
def __init__(self,scale_output=False):
super(DotProductSimilarity,self).__init__()
self.scale_output=scale_output
def forward(self,tensor_1,tensor_2):
result=(tensor_1*tensor_2).sum(dim=-1)
if(self.scale_output):
result/=math.sqrt(tensor_1.size(-1))
return result
input_1=torch.randn(3,5,requires_grad=True)
print(input_1)
input_2=torch.randn(3,5,requires_grad=True)
print(input_2)
con=ConineSimilarity()
CS=con(input_1,input_2)
print(CS)
dot=DotProductSimilarity()
DS=dot(input_1,input_2)
print(DS)
輸出：
tensor([[ 0.2148, 0.2645, 0.8717, -1.4637, 1.0932],
[-0.2975, -0.1585, 0.1567, -1.8238, -0.1437],
[-0.8243, -0.0170, -0.7533, -1.7643, 2.2571]], requires_grad=True)
tensor([[-1.8424, -1.4204, -1.5639, 0.2739, 0.7922],
[-0.3378, -2.3861, -1.7581, -0.8220, 0.3456],
[ 0.5536, 0.8324, -1.0632, -1.5567, 0.6374]], requires_grad=True)
tensor([-0.2780, 0.2843, 0.6595], grad_fn=<SumBackward2>)
tensor([-1.6696, 1.6526, 4.5157], grad_fn=<SumBackward2>)
餘弦相似度的大小固定在-1-1之間，但是點乘卻沒有。
3 雙線性相似度
計算公式如下：
b=x^T W y + b
class BiLinearSimilarity(nn.Module):
def __init__(self,tensor_1_dim,tensor_2_dim,activation=None):
super(BiLinearSimilarity,self).__init__()
self.weight_matrix=nn.Parameter(torch.Tensor(tensor_1_dim,tensor_2_dim))
self.bias=nn.Parameter(torch.Tensor(1))
self.activation=activation
self.reset_parameters()
def reset_parameters(self):
nn.init.xavier_uniform_(self.weight_matrix)
self.bias.data.fill_(0)
def forward(self, tensor_1,tensor_2):
intermediate=torch.matmul(tensor_1,self.weight_matrix)
result=(intermediate*tensor_2).sum(dim=-1)+self.bias
if self.activation is not None:
result=self.activation(result)
return result
input_1=torch.randn(3,5,requires_grad=True)
print(input_1)
input_2=torch.randn(3,5,requires_grad=True)
print(input_2)
con=ConineSimilarity()
CS=con(input_1,input_2)
print(CS)
dot=DotProductSimilarity()
DS=dot(input_1,input_2)
print(DS)
bilinear=BiLinearSimilarity(5,5)
BS=bilinear(input_1,input_2)
print(BS)
bilinear=BiLinearSimilarity(5,5)
BS=bilinear(input_1,input_2)
print(BS)
輸出：
tensor([[ 0.1465, -0.8367, 0.5901, -0.7421, 0.7775],
[ 0.2198, 1.5482, 1.5923, 1.0962, -1.6559],
[-1.8222, -0.7740, -0.4819, -1.8031, 1.6413]], requires_grad=True)
tensor([[ 0.0053, 2.1358, 0.9435, 0.2379, 0.0689],
[-0.3503, 1.3089, 0.5033, -2.2339, 1.7092],
[-0.0255, 1.0277, 1.4797, 0.2870, -0.4866]], requires_grad=True)
tensor([-0.3862, -0.2676, -0.4630], grad_fn=<SumBackward2>)
tensor([-1.3525, -2.5285, -2.7782], grad_fn=<SumBackward2>)
tensor([ 0.0766, 3.2140, -2.2844], grad_fn=<AddBackward0>)
從上可見，雙線性也並沒有取值範圍的限定。
以下是我根據自己理解上面的函數寫的公式，感覺跟官方一些的對不上。
4 歐式距離
input_1=torch.randn(3,5,requires_grad=True)
print(input_1)
input_2=torch.randn(3,5,requires_grad=True)
print(input_2)
con=ConineSimilarity()
CS=con(input_1,input_2)
print(CS)
dot=DotProductSimilarity()
DS=dot(input_1,input_2)
print(DS)
bilinear=BiLinearSimilarity(5,5)
BS=bilinear(input_1,input_2)
print(BS)
bilinear=BiLinearSimilarity(5,5)
BS=bilinear(input_1,input_2)
print(BS)
import torch.nn.functional as F
e=F.pairwise_distance(input_1,input_2)
print(e)
輸出：
tensor([[-1.0758, 1.2539, -1.4611, -0.5173, -0.3719],
[-0.3430, 1.0673, 0.3724, 0.2630, -0.0029],
[ 0.0570, 0.9101, 0.3809, 0.9970, 0.1919]], requires_grad=True)
tensor([[ 0.0319, -0.7283, -1.2269, 0.0922, 1.3412],
[-1.9286, 0.4859, -1.2347, 2.1949, -0.3295],
[-0.3811, 0.1348, -0.4247, -1.1917, 0.6632]], requires_grad=True)
tensor([ 0.0663, 0.3326, -0.5333], grad_fn=<SumBackward2>)
tensor([ 0.2986, 1.2986, -1.1216], grad_fn=<SumBackward2>)
tensor([ 0.4084, -0.2896, -0.4954], grad_fn=<AddBackward0>)
tensor([ 0.8389, 0.9224, -0.4486], grad_fn=<AddBackward0>)
tensor([2.9184, 3.0453, 2.5405], grad_fn=<NormBackward1>)
歐式距離輸出範圍也是不確定的。
5 皮爾遜相關係數
計算公式如下：
class PearsonCorrelation(nn.Module):
def forward(self,tensor_1,tensor_2):
x = tensor_1
y = tensor_2
vx = x - torch.mean(x)
vy = y - torch.mean(y)
cost = torch.sum(vx * vy) / (torch.sqrt(torch.sum(vx ** 2)) * torch.sqrt(torch.sum(vy ** 2)))
return cost
input_1=torch.randn(3,5,requires_grad=True)
print(input_1)
input_2=torch.randn(3,5,requires_grad=True)
print(input_2)
pearson=PearsonCorrelation()
PC=pearson(input_1,input_2)
print(PC)
輸出：
tensor([[-0.3064, 0.5541, 0.5944, 1.2654, 0.0281],
[ 1.0192, -0.8178, 0.2624, -1.4695, 0.3750],
[-1.7883, 0.5552, -0.1381, -0.6302, 0.6420]], requires_grad=True)
tensor([[ 0.8128, -0.3573, -0.8618, 1.2499, 0.5793],
[ 1.7729, 0.1705, -1.6051, -2.5970, 1.2015],
[-0.4328, 0.4294, 0.7364, 1.7886, 0.4203]], requires_grad=True)
tensor(0.4113, grad_fn=<DivBackward0>)
皮爾遜相關係數的值也限定在【-1,1】之間。
參考文獻：
常用的計算向量相似度的函數（pytorch版本）
Pytorch歐式距離euclidean distance實現
皮爾遜在pytorch的應用問題