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| author | Nicolas Courty <Nico@pc-mna-08.univ-ubs.fr> | 2017-08-28 14:41:09 +0200 |
|---|---|---|
| committer | Nicolas Courty <Nico@pc-mna-08.univ-ubs.fr> | 2017-08-28 14:41:09 +0200 |
| commit | 7ab9037f1e4a08439083d1bc5705be5ed2e9e10a (patch) | |
| tree | d8d5be164f7c45d67e68b4b4bb545e1b21b70e17 /ot/gromov.py | |
| parent | 7638d019b43e52d17600cac653939e7cd807478c (diff) | |
Gromov-Wasserstein distance
Diffstat (limited to 'ot/gromov.py')
| -rw-r--r-- | ot/gromov.py | 482 |
1 files changed, 482 insertions, 0 deletions
diff --git a/ot/gromov.py b/ot/gromov.py new file mode 100644 index 0000000..f3c62c9 --- /dev/null +++ b/ot/gromov.py @@ -0,0 +1,482 @@ +
+# -*- coding: utf-8 -*-
+"""
+Gromov-Wasserstein transport method
+===================================
+
+
+"""
+
+# Author: Erwan Vautier <erwan.vautier@gmail.com>
+# Nicolas Courty <ncourty@irisa.fr>
+#
+# License: MIT License
+
+import numpy as np
+
+from .bregman import sinkhorn
+from .utils import dist
+
+def square_loss(a,b):
+ """
+ Returns the value of L(a,b)=(1/2)*|a-b|^2
+ """
+
+ return (1/2)*(a-b)**2
+
+def kl_loss(a,b):
+ """
+ Returns the value of L(a,b)=a*log(a/b)-a+b
+ """
+
+ return a*np.log(a/b)-a+b
+
+def tensor_square_loss(C1,C2,T):
+ """
+ Returns the value of \mathcal{L}(C1,C2) \otimes T with the square loss
+
+ function as the loss function of Gromow-Wasserstein discrepancy.
+
+ Where :
+
+ C1 : Metric cost matrix in the source space
+ C2 : Metric cost matrix in the target space
+ T : A coupling between those two spaces
+
+ The square-loss function L(a,b)=(1/2)*|a-b|^2 is read as :
+ L(a,b) = f1(a)+f2(b)-h1(a)*h2(b) with :
+ f1(a)=(a^2)/2
+ f2(b)=(b^2)/2
+ h1(a)=a
+ h2(b)=b
+
+ Parameters
+ ----------
+ C1 : np.ndarray(ns,ns)
+ Metric cost matrix in the source space
+ C2 : np.ndarray(nt,nt)
+ Metric costfr matrix in the target space
+ T : np.ndarray(ns,nt)
+ Coupling between source and target spaces
+
+
+ Returns
+ -------
+ tens : (ns*nt) ndarray
+ \mathcal{L}(C1,C2) \otimes T tensor-matrix multiplication result
+
+
+ """
+
+
+ C1=np.asarray(C1,dtype=np.float64)
+ C2=np.asarray(C2,dtype=np.float64)
+ T=np.asarray(T,dtype=np.float64)
+
+
+ def f1(a):
+ return (a**2)/2
+
+ def f2(b):
+ return (b**2)/2
+
+ def h1(a):
+ return a
+
+ def h2(b):
+ return b
+
+ tens=-np.dot(h1(C1),T).dot(h2(C2).T)
+ tens=tens-tens.min()
+
+
+ return np.array(tens)
+
+
+def tensor_kl_loss(C1,C2,T):
+ """
+ Returns the value of \mathcal{L}(C1,C2) \otimes T with the square loss
+
+ function as the loss function of Gromow-Wasserstein discrepancy.
+
+ Where :
+
+ C1 : Metric cost matrix in the source space
+ C2 : Metric cost matrix in the target space
+ T : A coupling between those two spaces
+
+ The square-loss function L(a,b)=(1/2)*|a-b|^2 is read as :
+ L(a,b) = f1(a)+f2(b)-h1(a)*h2(b) with :
+ f1(a)=a*log(a)-a
+ f2(b)=b
+ h1(a)=a
+ h2(b)=log(b)
+
+ Parameters
+ ----------
+ C1 : np.ndarray(ns,ns)
+ Metric cost matrix in the source space
+ C2 : np.ndarray(nt,nt)
+ Metric costfr matrix in the target space
+ T : np.ndarray(ns,nt)
+ Coupling between source and target spaces
+
+
+ Returns
+ -------
+ tens : (ns*nt) ndarray
+ \mathcal{L}(C1,C2) \otimes T tensor-matrix multiplication result
+
+ References
+ ----------
+
+ .. [12] Peyré, Gabriel, Marco Cuturi, and Justin Solomon, "Gromov-Wasserstein averaging of kernel and distance matrices." International Conference on Machine Learning (ICML). 2016.
+
+ """
+
+
+ C1=np.asarray(C1,dtype=np.float64)
+ C2=np.asarray(C2,dtype=np.float64)
+ T=np.asarray(T,dtype=np.float64)
+
+
+ def f1(a):
+ return a*np.log(a+1e-15)-a
+
+ def f2(b):
+ return b
+
+ def h1(a):
+ return a
+
+ def h2(b):
+ return np.log(b+1e-15)
+
+ tens=-np.dot(h1(C1),T).dot(h2(C2).T)
+ tens=tens-tens.min()
+
+
+
+ return np.array(tens)
+
+def update_square_loss(p,lambdas,T,Cs):
+ """
+ Updates C according to the L2 Loss kernel with the S Ts couplings calculated at each iteration
+
+
+ Parameters
+ ----------
+ p : np.ndarray(N,)
+ weights in the targeted barycenter
+ lambdas : list of the S spaces' weights
+ T : list of S np.ndarray(ns,N)
+ the S Ts couplings calculated at each iteration
+ Cs : Cs : list of S np.ndarray(ns,ns)
+ Metric cost matrices
+
+ Returns
+ ----------
+ C updated
+
+
+ """
+ tmpsum=np.sum([lambdas[s]*np.dot(T[s].T,Cs[s]).dot(T[s]) for s in range(len(T))])
+ ppt=np.dot(p,p.T)
+
+ return(np.divide(tmpsum,ppt))
+
+def update_kl_loss(p,lambdas,T,Cs):
+ """
+ Updates C according to the KL Loss kernel with the S Ts couplings calculated at each iteration
+
+
+ Parameters
+ ----------
+ p : np.ndarray(N,)
+ weights in the targeted barycenter
+ lambdas : list of the S spaces' weights
+ T : list of S np.ndarray(ns,N)
+ the S Ts couplings calculated at each iteration
+ Cs : Cs : list of S np.ndarray(ns,ns)
+ Metric cost matrices
+
+ Returns
+ ----------
+ C updated
+
+
+ """
+ tmpsum=np.sum([lambdas[s]*np.dot(T[s].T,Cs[s]).dot(T[s]) for s in range(len(T))])
+ ppt=np.dot(p,p.T)
+
+ return(np.exp(np.divide(tmpsum,ppt)))
+
+
+def gromov_wasserstein(C1,C2,p,q,loss_fun,epsilon,numItermax = 1000, stopThr=1e-9,verbose=False, log=False):
+ """
+ Returns the gromov-wasserstein coupling between the two measured similarity matrices
+
+ (C1,p) and (C2,q)
+
+ The function solves the following optimization problem:
+
+ .. math::
+ \GW = arg\min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+
+ s.t. \GW 1 = p
+
+ \GW^T 1= q
+
+ \GW\geq 0
+
+ Where :
+
+ C1 : Metric cost matrix in the source space
+ C2 : Metric cost matrix in the target space
+ p : distribution in the source space
+ q : distribution in the target space
+ L : loss function to account for the misfit between the similarity matrices
+ H : entropy
+
+
+ Parameters
+ ----------
+ C1 : np.ndarray(ns,ns)
+ Metric cost matrix in the source space
+ C2 : np.ndarray(nt,nt)
+ Metric costfr matrix in the target space
+ p : np.ndarray(ns,)
+ distribution in the source space
+ q : np.ndarray(nt)
+ distribution in the target space
+ loss_fun : loss function used for the solver either 'square_loss' or 'kl_loss'
+ epsilon : float
+ Regularization term >0
+ numItermax : int, optional
+ Max number of iterations
+ stopThr : float, optional
+ Stop threshold on error (>0)
+ verbose : bool, optional
+ Print information along iterations
+ log : bool, optional
+ record log if True
+ forcing : np.ndarray(N,2)
+ list of forced couplings (where N is the number of forcing)
+
+ Returns
+ -------
+ T : coupling between the two spaces that minimizes :
+ \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+
+ """
+
+
+ C1=np.asarray(C1,dtype=np.float64)
+ C2=np.asarray(C2,dtype=np.float64)
+
+ T=np.dot(p,q.T) #Initialization
+
+ cpt = 0
+ err=1
+
+ while (err>stopThr and cpt<numItermax):
+
+ Tprev=T
+
+ if loss_fun=='square_loss':
+ tens=tensor_square_loss(C1,C2,T)
+
+ elif loss_fun=='kl_loss':
+ tens=tensor_kl_loss(C1,C2,T)
+
+ T=sinkhorn(p,q,tens,epsilon)
+
+ if cpt%10==0:
+ # we can speed up the process by checking for the error only all the 10th iterations
+ err=np.linalg.norm(T-Tprev)
+
+ if log:
+ log['err'].append(err)
+
+ if verbose:
+ if cpt%200 ==0:
+ print('{:5s}|{:12s}'.format('It.','Err')+'\n'+'-'*19)
+ print('{:5d}|{:8e}|'.format(cpt,err))
+
+ cpt=cpt+1
+
+ if log:
+ return T,log
+ else:
+ return T
+
+
+def gromov_wasserstein2(C1,C2,p,q,loss_fun,epsilon,numItermax = 1000, stopThr=1e-9,verbose=False, log=False):
+ """
+ Returns the gromov-wasserstein discrepancy between the two measured similarity matrices
+
+ (C1,p) and (C2,q)
+
+ The function solves the following optimization problem:
+
+ .. math::
+ \GW_Dist = \min_T \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+
+
+ Where :
+
+ C1 : Metric cost matrix in the source space
+ C2 : Metric cost matrix in the target space
+ p : distribution in the source space
+ q : distribution in the target space
+ L : loss function to account for the misfit between the similarity matrices
+ H : entropy
+
+
+ Parameters
+ ----------
+ C1 : np.ndarray(ns,ns)
+ Metric cost matrix in the source space
+ C2 : np.ndarray(nt,nt)
+ Metric costfr matrix in the target space
+ p : np.ndarray(ns,)
+ distribution in the source space
+ q : np.ndarray(nt)
+ distribution in the target space
+ loss_fun : loss function used for the solver either 'square_loss' or 'kl_loss'
+ epsilon : float
+ Regularization term >0
+ numItermax : int, optional
+ Max number of iterations
+ stopThr : float, optional
+ Stop threshold on error (>0)
+ verbose : bool, optional
+ Print information along iterations
+ log : bool, optional
+ record log if True
+ forcing : np.ndarray(N,2)
+ list of forced couplings (where N is the number of forcing)
+
+ Returns
+ -------
+ T : coupling between the two spaces that minimizes :
+ \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*T_{i,j}*T_{k,l}-\epsilon(H(T))
+
+ """
+
+ if log:
+ gw,logv=gromov_wasserstein(C1,C2,p,q,loss_fun,epsilon,numItermax,stopThr,verbose,log)
+ else:
+ gw=gromov_wasserstein(C1,C2,p,q,loss_fun,epsilon,numItermax,stopThr,verbose,log)
+
+ if loss_fun=='square_loss':
+ gw_dist=np.sum(gw*tensor_square_loss(C1,C2,gw))
+
+
+ elif loss_fun=='kl_loss':
+ gw_dist=np.sum(gw*tensor_kl_loss(C1,C2,gw))
+
+ if log:
+ return gw_dist, logv
+ else:
+ return gw_dist
+
+
+def gromov_barycenters(N,Cs,ps,p,lambdas,loss_fun,epsilon,numItermax = 1000, stopThr=1e-9, verbose=False, log=False):
+ """
+ Returns the gromov-wasserstein barycenters of S measured similarity matrices
+
+ (Cs)_{s=1}^{s=S}
+
+ The function solves the following optimization problem:
+
+ .. math::
+ C = argmin_C\in R^NxN \sum_s \lambda_s GW(C,Cs,p,ps)
+
+
+ Where :
+
+ Cs : metric cost matrix
+ ps : distribution
+
+ Parameters
+ ----------
+ N : Integer
+ Size of the targeted barycenter
+ Cs : list of S np.ndarray(ns,ns)
+ Metric cost matrices
+ ps : list of S np.ndarray(ns,)
+ sample weights in the S spaces
+ p : np.ndarray(N,)
+ weights in the targeted barycenter
+ lambdas : list of the S spaces' weights
+ L : tensor-matrix multiplication function based on specific loss function
+ update : function(p,lambdas,T,Cs) that updates C according to a specific Kernel
+ with the S Ts couplings calculated at each iteration
+ epsilon : float
+ Regularization term >0
+ numItermax : 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
+ -------
+ C : Similarity matrix in the barycenter space (permutated arbitrarily)
+
+ """
+
+
+ S=len(Cs)
+
+ Cs=[np.asarray(Cs[s],dtype=np.float64) for s in range(S)]
+ lambdas=np.asarray(lambdas,dtype=np.float64)
+
+ T=[0 for s in range(S)]
+
+ #Initialization of C : random SPD matrix
+ xalea=np.random.randn(N,2)
+ C=dist(xalea,xalea)
+ C/=C.max()
+
+ cpt=0
+ err=1
+
+ error=[]
+
+ while(err>stopThr and cpt<numItermax):
+
+ Cprev=C
+
+ T=[gromov_wasserstein(Cs[s],C,ps[s],p,loss_fun,epsilon,numItermax,1e-5,verbose,log) for s in range(S)]
+
+ if loss_fun=='square_loss':
+ C=update_square_loss(p,lambdas,T,Cs)
+
+ elif loss_fun=='kl_loss':
+ C=update_kl_loss(p,lambdas,T,Cs)
+
+ if cpt%10==0:
+ # we can speed up the process by checking for the error only all the 10th iterations
+ err=np.linalg.norm(C-Cprev)
+ error.append(err)
+
+ if log:
+ log['err'].append(err)
+
+ if verbose:
+ if cpt%200 ==0:
+ print('{:5s}|{:12s}'.format('It.','Err')+'\n'+'-'*19)
+ print('{:5d}|{:8e}|'.format(cpt,err))
+
+ cpt=cpt+1
+
+ return C
+
+
+
+
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