Function gcg

ot/optim.py:808–923

r""" Solve the general regularized OT problem with the generalized conditional gradient The function solves the following optimization problem: .. math:: \gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F + \mathrm{reg_1}\cdot\Omega(\gamma

(
    a,
    b,
    M,
    reg1,
    reg2,
    f,
    df,
    G0=None,
    numItermax=10,
    numInnerItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    verbose=False,
    log=False,
    **kwargs,
)

def gcg(
    a,
    b,
    M,
    reg1,
    reg2,
    f,
    df,
    G0=None,
    numItermax=10,
    numInnerItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    verbose=False,
    log=False,
    **kwargs,
):
    r"""
    Solve the general regularized OT problem with the generalized conditional gradient

        The function solves the following optimization problem:

    .. math::
        \gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F +
        \mathrm{reg_1}\cdot\Omega(\gamma) + \mathrm{reg_2}\cdot f(\gamma)

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

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

             \gamma &\geq 0

    where :

    - :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix
    - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
    - :math:`f` is the regularization term (and `df` is its gradient)
    - :math:`\mathbf{a}` and :math:`\mathbf{b}` are source and target weights (sum to 1)

    The algorithm used for solving the problem is the generalized conditional gradient as discussed in :ref:`[5, 7] <references-gcg>`


    Parameters
    ----------
    a : array-like, shape (ns,)
        samples weights in the source domain
    b : array-like, (nt,)
        samples in the target domain
    M : array-like, shape (ns, nt)
        loss matrix
    reg1 : float
        Entropic Regularization term >0
    reg2 : float
        Second Regularization term >0
    G0 : array-like, shape (ns, nt), optional
        initial guess (default is indep joint density)
    numItermax : int, optional
        Max number of iterations