Function weak_optimal_transport

ot/weak.py:16–130

r"""Solves the weak optimal transport problem between two empirical distributions .. math:: \gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2 s.t. \ \gamma \mathbf{1} = \mathbf

(
    Xa, Xb, a=None, b=None, verbose=False, log=False, G0=None, **kwargs
)

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 \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2

        s.t. \ \gamma \mathbf{1} = \mathbf{a}

             \gamma^T \mathbf{1} = \mathbf{b}

             \gamma \geq 0

    where :

    - :math:`X^a` and  :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))
    G0 : (ns,nt) array-like, float
        initial guess (default is indep joint density)
    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