Merge tag '0.8.2' into dfsg/latest

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+"""
+Weak optimal ransport solvers
+"""
+
+# Author: Remi Flamary <remi.flamary@polytehnique.edu>
+#
+# License: MIT License
+
+from .backend import get_backend
+from .optim import cg
+import numpy as np
+
+__all__ = ['weak_optimal_transport']
+
+
+def weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=None, **kwargs):
+    r"""Solves the weak optimal transport problem between two empirical distributions
+
+
+    .. math::
+        \gamma = \mathop{\arg \min}_\gamma \quad \|X_a-diag(1/a)\gammaX_b\|_F^2
+
+        s.t. \ \gamma \mathbf{1} = \mathbf{a}
+
+             \gamma^T \mathbf{1} = \mathbf{b}
+
+             \gamma \geq 0
+
+    where :
+
+    - :math:`X_a` :math:`X_b`  are the sample matrices.
+    - :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights
+
+
+    .. note:: This function is backend-compatible and will work on arrays
+        from all compatible backends. But the algorithm uses the C++ CPU backend
+        which can lead to copy overhead on GPU arrays.
+
+    Uses the conditional gradient algorithm to solve the problem proposed
+    in :ref:`[39] <references-weak>`.
+
+    Parameters
+    ----------
+    Xa : (ns,d) array-like, float
+        Source samples
+    Xb : (nt,d) array-like, float
+        Target samples
+    a : (ns,) array-like, float
+        Source histogram (uniform weight if empty list)
+    b : (nt,) array-like, float
+        Target histogram (uniform weight if empty list))
+    numItermax : int, optional
+        Max number of iterations
+    numItermaxEmd : int, optional
+        Max number of iterations for emd
+    stopThr : float, optional
+        Stop threshold on the relative variation (>0)
+    stopThr2 : float, optional
+        Stop threshold on the absolute variation (>0)
+    verbose : bool, optional
+        Print information along iterations
+    log : bool, optional
+        record log if True
+
+
+    Returns
+    -------
+    gamma: array-like, shape (ns, nt)
+        Optimal transportation matrix for the given
+        parameters
+    log: dict, optional
+        If input log is true, a dictionary containing the
+        cost and dual variables and exit status
+
+
+    .. _references-weak:
+    References
+    ----------
+    .. [39] Gozlan, N., Roberto, C., Samson, P. M., & Tetali, P. (2017).
+            Kantorovich duality for general transport costs and applications.
+            Journal of Functional Analysis, 273(11), 3327-3405.
+
+    See Also
+    --------
+    ot.bregman.sinkhorn : Entropic regularized OT
+    ot.optim.cg : General regularized OT
+    """
+
+    nx = get_backend(Xa, Xb)
+
+    Xa2 = nx.to_numpy(Xa)
+    Xb2 = nx.to_numpy(Xb)
+
+    if a is None:
+        a2 = np.ones((Xa.shape[0])) / Xa.shape[0]
+    else:
+        a2 = nx.to_numpy(a)
+    if b is None:
+        b2 = np.ones((Xb.shape[0])) / Xb.shape[0]
+    else:
+        b2 = nx.to_numpy(b)
+
+    # init uniform
+    if G0 is None:
+        T0 = a2[:, None] * b2[None, :]
+    else:
+        T0 = nx.to_numpy(G0)
+
+    # weak OT loss
+    def f(T):
+        return np.dot(a2, np.sum((Xa2 - np.dot(T, Xb2) / a2[:, None])**2, 1))
+
+    # weak OT gradient
+    def df(T):
+        return -2 * np.dot(Xa2 - np.dot(T, Xb2) / a2[:, None], Xb2.T)
+
+    # solve with conditional gradient and return solution
+    if log:
+        res, log = cg(a2, b2, 0, 1, f, df, T0, log=log, verbose=verbose, **kwargs)
+        log['u'] = nx.from_numpy(log['u'], type_as=Xa)
+        log['v'] = nx.from_numpy(log['v'], type_as=Xb)
+        return nx.from_numpy(res, type_as=Xa), log
+    else:
+        return nx.from_numpy(cg(a2, b2, 0, 1, f, df, T0, log=log, verbose=verbose, **kwargs), type_as=Xa)