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# -*- coding: utf-8 -*-
"""
Rewrite ot.bregman.sinkhorn in Python Optimal Transport (https://pythonot.github.io/_modules/ot/bregman.html#sinkhorn)
using pytorch operations.
Bregman projections for regularized OT (Sinkhorn distance).
"""
import torch
M_EPS = 1e-16
def sinkhorn(a, b, C, reg=1e-1, method='sinkhorn', maxIter=1000, tau=1e3,
stopThr=1e-9, verbose=False, log=True, warm_start=None, eval_freq=10, print_freq=200, **kwargs):
"""
Solve the entropic regularization optimal transport
The input should be PyTorch tensors
The function solves the following optimization problem:
.. math::
\gamma = arg\min_\gamma <\gamma,C>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = a
\gamma^T 1= b
\gamma\geq 0
where :
- C is the (ns,nt) metric cost matrix
- :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
- a and b are target and source measures (sum to 1)
The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [1].
Parameters
----------
a : torch.tensor (na,)
samples measure in the target domain
b : torch.tensor (nb,)
samples in the source domain
C : torch.tensor (na,nb)
loss matrix
reg : float
Regularization term > 0
method : str
method used for the solver either 'sinkhorn', 'greenkhorn', 'sinkhorn_stabilized' or
'sinkhorn_epsilon_scaling', see those function for specific parameters
maxIter : int, optional
Max number of iterations
stopThr : float, optional
Stop threshol on error ( > 0 )
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
Returns
-------
gamma : (na x nb) torch.tensor
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
References
----------
[1] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013
See Also
--------
"""
if method.lower() == 'sinkhorn':
return sinkhorn_knopp(a, b, C, reg, maxIter=maxIter,
stopThr=stopThr, verbose=verbose, log=log,
warm_start=warm_start, eval_freq=eval_freq, print_freq=print_freq,
**kwargs)
elif method.lower() == 'sinkhorn_stabilized':
return sinkhorn_stabilized(a, b, C, reg, maxIter=maxIter, tau=tau,
stopThr=stopThr, verbose=verbose, log=log,
warm_start=warm_start, eval_freq=eval_freq, print_freq=print_freq,
**kwargs)
elif method.lower() == 'sinkhorn_epsilon_scaling':
return sinkhorn_epsilon_scaling(a, b, C, reg,
maxIter=maxIter, maxInnerIter=100, tau=tau,
scaling_base=0.75, scaling_coef=None, stopThr=stopThr,
verbose=False, log=log, warm_start=warm_start, eval_freq=eval_freq,
print_freq=print_freq, **kwargs)
else:
raise ValueError("Unknown method '%s'." % method)
def sinkhorn_knopp(a, b, C, reg=1e-1, maxIter=1000, stopThr=1e-9,
verbose=False, log=False, warm_start=None, eval_freq=10, print_freq=200, **kwargs):
"""
Solve the entropic regularization optimal transport
The input should be PyTorch tensors
The function solves the following optimization problem:
.. math::
\gamma = arg\min_\gamma <\gamma,C>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = a
\gamma^T 1= b
\gamma\geq 0
where :
- C is the (ns,nt) metric cost matrix
- :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
- a and b are target and source measures (sum to 1)
The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [1].
Parameters
----------
a : torch.tensor (na,)
samples measure in the target domain
b : torch.tensor (nb,)
samples in the source domain
C : torch.tensor (na,nb)
loss matrix
reg : float
Regularization term > 0
maxIter : int, optional
Max number of iterations
stopThr : float, optional
Stop threshol on error ( > 0 )
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
Returns
-------
gamma : (na x nb) torch.tensor
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
References
----------
[1] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013
See Also
--------
"""
device = a.device
na, nb = C.shape
assert na >= 1 and nb >= 1, 'C needs to be 2d'
assert na == a.shape[0] and nb == b.shape[0], "Shape of a or b does't match that of C"
assert reg > 0, 'reg should be greater than 0'
assert a.min() >= 0. and b.min() >= 0., 'Elements in a or b less than 0'
if log:
log = {'err': []}
if warm_start is not None:
u = warm_start['u']
v = warm_start['v']
else:
u = torch.ones(na, dtype=a.dtype).to(device) / na
v = torch.ones(nb, dtype=b.dtype).to(device) / nb
K = torch.empty(C.shape, dtype=C.dtype).to(device)
torch.div(C, -reg, out=K)
torch.exp(K, out=K)
b_hat = torch.empty(b.shape, dtype=C.dtype).to(device)
it = 1
err = 1
# allocate memory beforehand
KTu = torch.empty(v.shape, dtype=v.dtype).to(device)
Kv = torch.empty(u.shape, dtype=u.dtype).to(device)
while (err > stopThr and it <= maxIter):
upre, vpre = u, v
torch.matmul(u, K, out=KTu)
v = torch.div(b, KTu + M_EPS)
torch.matmul(K, v, out=Kv)
u = torch.div(a, Kv + M_EPS)
if torch.any(torch.isnan(u)) or torch.any(torch.isnan(v)) or \
torch.any(torch.isinf(u)) or torch.any(torch.isinf(v)):
print('Warning: numerical errors at iteration', it)
u, v = upre, vpre
break
if log and it % eval_freq == 0:
# we can speed up the process by checking for the error only all
# the eval_freq iterations
# below is equivalent to:
# b_hat = torch.sum(u.reshape(-1, 1) * K * v.reshape(1, -1), 0)
# but with more memory efficient
b_hat = torch.matmul(u, K) * v
err = (b - b_hat).pow(2).sum().item()
# err = (b - b_hat).abs().sum().item()
log['err'].append(err)
if verbose and it % print_freq == 0:
print('iteration {:5d}, constraint error {:5e}'.format(it, err))
it += 1
if log:
log['u'] = u
log['v'] = v
log['alpha'] = reg * torch.log(u + M_EPS)
log['beta'] = reg * torch.log(v + M_EPS)
# transport plan
P = u.reshape(-1, 1) * K * v.reshape(1, -1)
if log:
return P, log
else:
return P
def sinkhorn_stabilized(a, b, C, reg=1e-1, maxIter=1000, tau=1e3, stopThr=1e-9,
verbose=False, log=False, warm_start=None, eval_freq=10, print_freq=200, **kwargs):
"""
Solve the entropic regularization OT problem with log stabilization
The function solves the following optimization problem:
.. math::
\gamma = arg\min_\gamma <\gamma,C>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = a
\gamma^T 1= b
\gamma\geq 0
where :
- C is the (ns,nt) metric cost matrix
- :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
- a and b are target and source measures (sum to 1)
The algorithm used for solving the problem is the Sinkhorn-Knopp matrix scaling algorithm as proposed in [1]
but with the log stabilization proposed in [3] an defined in [2] (Algo 3.1)
Parameters
----------
a : torch.tensor (na,)
samples measure in the target domain
b : torch.tensor (nb,)
samples in the source domain
C : torch.tensor (na,nb)
loss matrix
reg : float
Regularization term > 0
tau : float
thershold for max value in u or v for log scaling
maxIter : int, optional
Max number of iterations
stopThr : float, optional
Stop threshol on error ( > 0 )
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
Returns
-------
gamma : (na x nb) torch.tensor
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
References
----------
[1] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013
[2] Bernhard Schmitzer. Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. SIAM Journal on Scientific Computing, 2019
[3] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
See Also
--------
"""
device = a.device
na, nb = C.shape
assert na >= 1 and nb >= 1, 'C needs to be 2d'
assert na == a.shape[0] and nb == b.shape[0], "Shape of a or b does't match that of C"
assert reg > 0, 'reg should be greater than 0'
assert a.min() >= 0. and b.min() >= 0., 'Elements in a or b less than 0'
if log:
log = {'err': []}
if warm_start is not None:
alpha = warm_start['alpha']
beta = warm_start['beta']
else:
alpha = torch.zeros(na, dtype=a.dtype).to(device)
beta = torch.zeros(nb, dtype=b.dtype).to(device)
u = torch.ones(na, dtype=a.dtype).to(device) / na
v = torch.ones(nb, dtype=b.dtype).to(device) / nb
def update_K(alpha, beta):
"""log space computation"""
"""memory efficient"""
torch.add(alpha.reshape(-1, 1), beta.reshape(1, -1), out=K)
torch.add(K, -C, out=K)
torch.div(K, reg, out=K)
torch.exp(K, out=K)
def update_P(alpha, beta, u, v, ab_updated=False):
"""log space P (gamma) computation"""
torch.add(alpha.reshape(-1, 1), beta.reshape(1, -1), out=P)
torch.add(P, -C, out=P)
torch.div(P, reg, out=P)
if not ab_updated:
torch.add(P, torch.log(u + M_EPS).reshape(-1, 1), out=P)
torch.add(P, torch.log(v + M_EPS).reshape(1, -1), out=P)
torch.exp(P, out=P)
K = torch.empty(C.shape, dtype=C.dtype).to(device)
update_K(alpha, beta)
b_hat = torch.empty(b.shape, dtype=C.dtype).to(device)
it = 1
err = 1
ab_updated = False
# allocate memory beforehand
KTu = torch.empty(v.shape, dtype=v.dtype).to(device)
Kv = torch.empty(u.shape, dtype=u.dtype).to(device)
P = torch.empty(C.shape, dtype=C.dtype).to(device)
while (err > stopThr and it <= maxIter):
upre, vpre = u, v
torch.matmul(u, K, out=KTu)
v = torch.div(b, KTu + M_EPS)
torch.matmul(K, v, out=Kv)
u = torch.div(a, Kv + M_EPS)
ab_updated = False
# remove numerical problems and store them in K
if u.abs().sum() > tau or v.abs().sum() > tau:
alpha += reg * torch.log(u + M_EPS)
beta += reg * torch.log(v + M_EPS)
u.fill_(1. / na)
v.fill_(1. / nb)
update_K(alpha, beta)
ab_updated = True
if log and it % eval_freq == 0:
# we can speed up the process by checking for the error only all
# the eval_freq iterations
update_P(alpha, beta, u, v, ab_updated)
b_hat = torch.sum(P, 0)
err = (b - b_hat).pow(2).sum().item()
log['err'].append(err)
if verbose and it % print_freq == 0:
print('iteration {:5d}, constraint error {:5e}'.format(it, err))
it += 1
if log:
log['u'] = u
log['v'] = v
log['alpha'] = alpha + reg * torch.log(u + M_EPS)
log['beta'] = beta + reg * torch.log(v + M_EPS)
# transport plan
update_P(alpha, beta, u, v, False)
if log:
return P, log
else:
return P
def sinkhorn_epsilon_scaling(a, b, C, reg=1e-1, maxIter=100, maxInnerIter=100, tau=1e3, scaling_base=0.75,
scaling_coef=None, stopThr=1e-9, verbose=False, log=False, warm_start=None, eval_freq=10,
print_freq=200, **kwargs):
"""
Solve the entropic regularization OT problem with log stabilization
The function solves the following optimization problem:
.. math::
\gamma = arg\min_\gamma <\gamma,C>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = a
\gamma^T 1= b
\gamma\geq 0
where :
- C is the (ns,nt) metric cost matrix
- :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
- a and b are target and source measures (sum to 1)
The algorithm used for solving the problem is the Sinkhorn-Knopp matrix
scaling algorithm as proposed in [1] but with the log stabilization
proposed in [3] and the log scaling proposed in [2] algorithm 3.2
Parameters
----------
a : torch.tensor (na,)
samples measure in the target domain
b : torch.tensor (nb,)
samples in the source domain
C : torch.tensor (na,nb)
loss matrix
reg : float
Regularization term > 0
tau : float
thershold for max value in u or v for log scaling
maxIter : int, optional
Max number of iterations
stopThr : float, optional
Stop threshol on error ( > 0 )
verbose : bool, optional
Print information along iterations
log : bool, optional
record log if True
Returns
-------
gamma : (na x nb) torch.tensor
Optimal transportation matrix for the given parameters
log : dict
log dictionary return only if log==True in parameters
References
----------
[1] M. Cuturi, Sinkhorn Distances : Lightspeed Computation of Optimal Transport, Advances in Neural Information Processing Systems (NIPS) 26, 2013
[2] Bernhard Schmitzer. Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. SIAM Journal on Scientific Computing, 2019
[3] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.
See Also
--------
"""
na, nb = C.shape
assert na >= 1 and nb >= 1, 'C needs to be 2d'
assert na == a.shape[0] and nb == b.shape[0], "Shape of a or b does't match that of C"
assert reg > 0, 'reg should be greater than 0'
assert a.min() >= 0. and b.min() >= 0., 'Elements in a or b less than 0'
def get_reg(it, reg, pre_reg):
if it == 1:
return scaling_coef
else:
if (pre_reg - reg) * scaling_base < M_EPS:
return reg
else:
return (pre_reg - reg) * scaling_base + reg
if scaling_coef is None:
scaling_coef = C.max() + reg
it = 1
err = 1
running_reg = scaling_coef
if log:
log = {'err': []}
warm_start = None
while (err > stopThr and it <= maxIter):
running_reg = get_reg(it, reg, running_reg)
P, _log = sinkhorn_stabilized(a, b, C, running_reg, maxIter=maxInnerIter, tau=tau,
stopThr=stopThr, verbose=False, log=True,
warm_start=warm_start, eval_freq=eval_freq, print_freq=print_freq,
**kwargs)
warm_start = {}
warm_start['alpha'] = _log['alpha']
warm_start['beta'] = _log['beta']
primal_val = (C * P).sum() + reg * (P * torch.log(P)).sum() - reg * P.sum()
dual_val = (_log['alpha'] * a).sum() + (_log['beta'] * b).sum() - reg * P.sum()
err = primal_val - dual_val
log['err'].append(err)
if verbose and it % print_freq == 0:
print('iteration {:5d}, constraint error {:5e}'.format(it, err))
it += 1
if log:
log['alpha'] = _log['alpha']
log['beta'] = _log['beta']
return P, log
else:
return P
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