【算法】最大均值差异(Maximum Mean Discrepancy, MMD)损失函数原理与python代码

MMD介绍
MMD(最大均值差异)是迁移学习,尤其是Domain adaptation (域适应)中使用最广泛(目前)的一种损失函数,主要用来度量两个不同但相关的分布的距离。两个分布的距离定义为:

python代码样例:

import torch

def guassian_kernel(source, target, kernel_mul=2.0, kernel_num=5, fix_sigma=None):
    '''
    将源域数据和目标域数据转化为核矩阵,即上文中的K
    Params:
	    source: 源域数据(n * len(x))
	    target: 目标域数据(m * len(y))
	    kernel_mul:
	    kernel_num: 取不同高斯核的数量
	    fix_sigma: 不同高斯核的sigma值
	Return:
		sum(kernel_val): 多个核矩阵之和
    '''
    n_samples = int(source.size()[0])+int(target.size()[0])# 求矩阵的行数,一般source和target的尺度是一样的,这样便于计算
    total = torch.cat([source, target], dim=0)#将source,target按列方向合并
    #将total复制(n+m)份
    total0 = total.unsqueeze(0).expand(int(total.size(0)), int(total.size(0)), int(total.size(1)))
    #将total的每一行都复制成(n+m)行,即每个数据都扩展成(n+m)份
    total1 = total.unsqueeze(1).expand(int(total.size(0)), int(total.size(0)), int(total.size(1)))
    #求任意两个数据之间的和,得到的矩阵中坐标(i,j)代表total中第i行数据和第j行数据之间的l2 distance(i==j时为0)
    L2_distance = ((total0-total1)**2).sum(2)
    #调整高斯核函数的sigma值
    if fix_sigma:
        bandwidth = fix_sigma
    else:
        bandwidth = torch.sum(L2_distance.data) / (n_samples**2-n_samples)
    #以fix_sigma为中值,以kernel_mul为倍数取kernel_num个bandwidth值(比如fix_sigma为1时,得到[0.25,0.5,1,2,4]
    bandwidth /= kernel_mul ** (kernel_num // 2)
    bandwidth_list = [bandwidth * (kernel_mul**i) for i in range(kernel_num)]
    #高斯核函数的数学表达式
    kernel_val = [torch.exp(-L2_distance / bandwidth_temp) for bandwidth_temp in bandwidth_list]
    #得到最终的核矩阵
    return sum(kernel_val)#/len(kernel_val)

def mmd_rbf(source, target, kernel_mul=2.0, kernel_num=5, fix_sigma=None):
    '''
    计算源域数据和目标域数据的MMD距离
    Params:
	    source: 源域数据(n * len(x))
	    target: 目标域数据(m * len(y))
	    kernel_mul:
	    kernel_num: 取不同高斯核的数量
	    fix_sigma: 不同高斯核的sigma值
	Return:
		loss: MMD loss
    '''
    batch_size = int(source.size()[0])#一般默认为源域和目标域的batchsize相同
    kernels = guassian_kernel(source, target,
        kernel_mul=kernel_mul, kernel_num=kernel_num, fix_sigma=fix_sigma)
    #根据式(3)将核矩阵分成4部分
    XX = kernels[:batch_size, :batch_size]
    YY = kernels[batch_size:, batch_size:]
    XY = kernels[:batch_size, batch_size:]
    YX = kernels[batch_size:, :batch_size]
    loss = torch.mean(XX + YY - XY -YX)
    return loss#因为一般都是n==m,所以L矩阵一般不加入计算


参考文献:

https://blog.csdn.net/a529975125/article/details/81176029

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