sinkhorn算法
时间: 2023-09-12 17:11:08 浏览: 373
Sinkhorn算法是一种用于解决正方形的非负矩阵的最优传输问题的迭代算法。它可以用于求解两个分布之间的最小距离,也被广泛应用于图像处理、机器学习和计算机视觉等领域。
该算法是基于Kantorovich-Rubinstein对偶性的理论基础上发展而来的。它通过迭代更新行和列的缩放因子,使得矩阵中每一行和每一列之和满足给定的边际约束。具体来说,算法迭代地将行和列进行正规化,并且在每次迭代中保持矩阵的行和列和不变。通过多次迭代,矩阵逐渐收敛到满足边际约束的最优解。
Sinkhorn算法具有高效性和收敛性,并且在处理大规模问题时具有较好的性能。它已经被广泛应用于计算机视觉领域中的图像配准、图像生成和图像分类等任务中。同时,Sinkhorn算法也被用于解决其他领域中的最小距离问题,如自然语言处理和经济学等领域。
相关问题
生成sinkhorn算法代码
抱歉,作为AI语言模型,我无法直接生成代码。不过,以下是Python中使用sinkhorn算法的示例代码,供您参考:
```python
import numpy as np
def sinkhorn(K, r, c, reg, numItermax=1000, stopThr=1e-9):
"""
Solves the entropic regularization optimal transport problem and returns the OT matrix
The function solves the following optimization problem:
.. math::
\gamma = arg\min_\gamma <\gamma,K>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = \alpha
\gamma^T 1= \beta
\gamma\geq 0
where :
- K: loss matrix NxM
- r: row margins (sums) len(N)
- c: column margins (sums) len(M)
- reg: regularization term >0
- gamma: transport plan NxM
- <.,.>_F : Frobenius dot product, ie <A,B>_F = \sum_i\sum_j A_{i,j}B_{i,j}
- \Omega : entropy, ie \Omega(\gamma)=\sum_{i,j} \gamma_{i,j}(\log(\gamma_{i,j})-1)
The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [2]_
Parameters
----------
K : ndarray, shape (n_samples, n_features)
Ground distance matrix between the two point sets.
r : ndarray, shape (n_samples,)
Marginal of the first point set.
c : ndarray, shape (n_samples,)
Marginal of the second point set.
reg : float
Regularization term. The higher reg, the more sparse the solution.
numItermax : int, optional
Max number of iterations
stopThr : float, optional
Stop threshol on error (>0)
Returns
-------
gamma : ndarray, shape (n_samples, n_features)
Optimal transportation matrix for the given parameters
"""
assert len(r) == K.shape[0], "nb of rows in K and len(r) do not match"
assert len(c) == K.shape[1], "nb of cols in K and len(c) do not match"
r = np.asarray(r, dtype=np.float64)
c = np.asarray(c, dtype=np.float64)
K = np.asarray(K, dtype=np.float64)
# init data
Nini = len(r)
Nfin = len(c)
u = np.ones(Nini) / Nini
v = np.ones(Nfin) / Nfin
# print(reg)
Kp = (np.exp(-reg * K))
Ktransposepy = np.dot(Kp.transpose(), y)
cpt = 0
err = 1
while (err > stopThr and cpt < numItermax):
uprev = u
vprev = v
KtransposeU = np.dot(Kp, u)
v = c / KtransposeU
u = r / Ktransposepy
if (np.abs(u).max() > 1e9 or np.abs(v).max() > 1e9):
# to avoid numerical problem
# (we need to homogenize the matrix),
# better to restart
print('Warning: numerically unstable')
reg = reg * 10
u = np.ones(Nini) / Nini
v = np.ones(Nfin) / Nfin
Kp = (np.exp(-reg * K))
if cpt % 10 == 0:
# we can speed up the process by checking for the error only all
# the 10th iterations
transp = u.reshape((Nini, 1)) * Kp * v.reshape((1, Nfin))
err = np.linalg.norm(np.sum(transp, axis=1) - r) + \
np.linalg.norm(np.sum(transp, axis=0) - c)
# print("err=", err, "stab. coef=", np.sum(Kp * np.log(Kp / K)))
cpt += 1
return u.reshape((Nini, 1)) * Kp * v.reshape((1, Nfin))
```
这里使用了numpy库和一些数学运算符和函数,包括矩阵乘法和范数计算。具体实现可根据您的需求进行修改。
sinkhorn算法数学原理
Sinkhorn算法是一种用于解决凸优化问题的算法,它基于一种称为Sinkhorn-Knopp算子的迭代方法。该算法主要用于解决带有矩阵约束的优化问题,例如最小化两个矩阵之间的距离或最大化矩阵的熵。该算法的核心思想是通过交替地对矩阵的行和列进行缩放,来逐步逼近最优解。