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# -*- coding: utf-8 -*-
"""
Created on Fri Oct 21 09:40:21 2016
@author: rflamary
"""
import numpy as np
def sinkhorn(a,b, M, reg,numItermax = 1000,stopThr=1e-9):
"""
Solve the optimal transport problem (OT)
.. math::
\gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\gamma)
s.t. \gamma 1 = a
\gamma^T 1= b
\gamma\geq 0
where :
- M is the metric cost matrix
- Omega is the entropic regularization term
- a and b are the sample weights
Parameters
----------
a : (ns,) ndarray
samples in the source domain
b : (nt,) ndarray
samples in the target domain
M : (ns,nt) ndarray
loss matrix
reg: float()
Regularization term >0
Returns
-------
gamma: (ns x nt) ndarray
Optimal transportation matrix for the given parameters
"""
# init data
Nini = len(a)
Nfin = len(b)
cpt = 0
# we assume that no distances are null except those of the diagonal of distances
u = np.ones(Nini)/Nini
v = np.ones(Nfin)/Nfin
uprev=np.zeros(Nini)
vprev=np.zeros(Nini)
#print reg
K = np.exp(-M/reg)
#print np.min(K)
Kp = np.dot(np.diag(1/a),K)
transp = K
cpt = 0
err=1
while (err>stopThr and cpt<numItermax):
if np.any(np.dot(K.T,u)==0) or np.any(np.isnan(u)) or np.any(np.isnan(v)):
# we have reached the machine precision
# come back to previous solution and quit loop
print('Warning: numerical errrors')
if cpt!=0:
u = uprev
v = vprev
break
uprev = u
vprev = v
v = np.divide(b,np.dot(K.T,u))
u = 1./np.dot(Kp,v)
if cpt%10==0:
# we can speed up the process by checking for the error only all the 10th iterations
transp = np.dot(np.diag(u),np.dot(K,np.diag(v)))
err = np.linalg.norm((np.sum(transp,axis=0)-b))**2
cpt = cpt +1
#print 'err=',err,' cpt=',cpt
return np.dot(np.diag(u),np.dot(K,np.diag(v)))
def geometricBar(weights,alldistribT):
assert(len(weights)==alldistribT.shape[1])
return np.exp(np.dot(np.log(alldistribT),weights.T))
def geometricMean(alldistribT):
return np.exp(np.mean(np.log(alldistribT),axis=1))
def projR(gamma,p):
#return np.dot(np.diag(p/np.maximum(np.sum(gamma,axis=1),1e-10)),gamma)
return np.multiply(gamma.T,p/np.maximum(np.sum(gamma,axis=1),1e-10)).T
def projC(gamma,q):
#return (np.dot(np.diag(q/np.maximum(np.sum(gamma,axis=0),1e-10)),gamma.T)).T
return np.multiply(gamma,q/np.maximum(np.sum(gamma,axis=0),1e-10))
def barycenter(A,M,reg, weights=None, numItermax = 1000, tol_error=1e-4,log=dict()):
"""Compute the Regularizzed wassersteien barycenter of distributions A"""
if weights is None:
weights=np.ones(A.shape[1])/A.shape[1]
else:
assert(len(weights)==A.shape[1])
#compute Mmax once for all
#M = M/np.median(M) # suggested by G. Peyre
K = np.exp(-M/reg)
cpt = 0
err=1
UKv=np.dot(K,np.divide(A.T,np.sum(K,axis=0)).T)
u = (geometricMean(UKv)/UKv.T).T
log['niter']=0
log['all_err']=[]
while (err>tol_error and cpt<numItermax):
cpt = cpt +1
UKv=u*np.dot(K,np.divide(A,np.dot(K,u)))
u = (u.T*geometricBar(weights,UKv)).T/UKv
if cpt%10==1:
err=np.sum(np.std(UKv,axis=1))
log['all_err'].append(err)
log['niter']=cpt
return geometricBar(weights,UKv),log
def unmix(distrib,D,M,M0,h0,reg,reg0,alpha,numItermax = 1000, tol_error=1e-3,log=dict()):
"""
distrib : distribution to unmix
D : Dictionnary
M : Metric matrix in the space of the distributions to unmix
M0 : Metric matrix in the space of the 'abundance values' to solve for
h0 : prior on solution (generally uniform distribution)
reg,reg0 : transport regularizations
alpha : how much should we trust the prior ? ([0,1])
"""
#M = M/np.median(M)
K = np.exp(-M/reg)
#M0 = M0/np.median(M0)
K0 = np.exp(-M0/reg0)
old = h0
err=1
cpt=0
#log = {'niter':0, 'all_err':[]}
log['niter']=0
log['all_err']=[]
while (err>tol_error and cpt<numItermax):
K = projC(K,distrib)
K0 = projC(K0,h0)
new = np.sum(K0,axis=1)
inv_new = np.dot(D,new) # we recombine the current selection from dictionnary
other = np.sum(K,axis=1)
delta = np.exp(alpha*np.log(other)+(1-alpha)*np.log(inv_new)) # geometric interpolation
K = projR(K,delta)
K0 = np.dot(np.diag(np.dot(D.T,delta/inv_new)),K0)
err=np.linalg.norm(np.sum(K0,axis=1)-old)
old = new
log['all_err'].append(err)
cpt = cpt+1
log['niter']=cpt
return np.sum(K0,axis=1),log
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