doc linear mapping estimation

Author: rflamary

ot/da.py

@@ -124,7 +124,80 @@ def sinkhorn_lpl1_mm(a,labels_a, b, M, reg, eta=0.1,numItermax = 10,numInnerIter
     return transp
 
 def joint_OT_mapping_linear(xs,xt,mu=1,eta=0.001,bias=False,verbose=False,verbose2=False,numItermax = 100,numInnerItermax = 10,stopInnerThr=1e-6,stopThr=1e-5,log=False,**kwargs):
-    """Joint Ot and mapping estimation (uniform weights and )
+    """Joint OT and linear mapping estimation as proposed in [8]
+
+    The function solves the following optimization problem:
+
+    .. math::
+        \min_{\gamma,L}\quad \|L(X_s) -n_s\gamma X_t\|^2_F + \mu<\gamma,M>_F + \eta  \|L -I\|^2_F
+
+        s.t. \gamma 1 = a
+
+             \gamma^T 1= b
+
+             \gamma\geq 0
+    where :
+
+    - M is the (ns,nt) squared euclidean cost matrix between samples in Xs and Xt (scaled by ns)
+    - :math:`L` is a dxd linear operator that approximates the barycentric mapping
+    - :math:`I` is the identity matrix (neutral linear mapping)
+    - a and b are uniform source and target weights
+
+    The problem consist in solving jointly an optimal transport matrix
+    :math:`\gamma` and a linear mapping that fits the barycentric mapping
+    :math:`n_s\gamma X_t`
+
+    The algorithm used for solving the problem is the block coordinate
+    descent that alternates between updates of G (using conditionnal gradient)
+    abd the update of L using a classical least square solver.
+
+
+    Parameters
+    ----------
+    xs : np.ndarray (ns,d)
+        samples in the source domain
+    xt : np.ndarray (nt,d)
+        samples in the target domain
+    mu: float,optional
+        Weight for the linear OT loss (>0)
+    eta: float, optional
+        Regularization term  for the linear mapping L (>0)
+    bias: bool,optional
+        Estimate linear mapping with constant bias
+    numItermax: int, optional
+        Max number of BCD iterations
+    stopThr: float, optional
+        Stop threshold on relative loss decrease (>0)
+    numInnerItermax: int, optional
+        Max number of iterations (inner CG solver)
+    stopInnerThr: float, optional
+        Stop threshold on error (inner CG solver) (>0)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+
+
+    Returns
+    -------
+    gamma: (ns x nt) ndarray
+        Optimal transportation matrix for the given parameters
+    L: (d x d) ndarray
+        Linear mapping matrix (d+1 x d if bias)
+    log: dict
+        log dictionary return only if log==True in parameters
+
+
+    References
+    ----------
+
+    .. [8] M. Perrot, N. Courty, R. Flamary, A. Habrard, "Mapping estimation for discrete optimal transport", Neural Information Processing Systems (NIPS), 2016.
+
+    See Also
+    --------
+    ot.lp.emd : Unregularized OT
+    ot.optim.cg : General regularized OT
+
     """
 
     ns,nt,d=xs.shape[0],xt.shape[0],xt.shape[1]