Function semirelaxed_cg

ot/optim.py:541–666

r""" Solve the general regularized and semi-relaxed OT problem with conditional gradient The function solves the following optimization problem: .. math:: \gamma = \mathop{\arg \min}_\gamma \quad \langle \gamma, \mathbf{M} \rangle_F + \mathrm{reg} \cdot f(\gamma)

(
    a,
    b,
    M,
    reg,
    f,
    df,
    G0=None,
    line_search=None,
    numItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    verbose=False,
    log=False,
    nx=None,
    **kwargs,
)

def semirelaxed_cg(
    a,
    b,
    M,
    reg,
    f,
    df,
    G0=None,
    line_search=None,
    numItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    verbose=False,
    log=False,
    nx=None,
    **kwargs,
):
    r"""
    Solve the general regularized and semi-relaxed OT problem with conditional gradient

        The function solves the following optimization problem:

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

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

             \gamma &\geq 0

    where :

    - :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix
    - :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 conditional gradient as discussed in :ref:`[1] <references-cg>`


    Parameters
    ----------
    a : array-like, shape (ns,)
        samples weights in the source domain
    b : array-like, shape (nt,)
        currently estimated samples weights in the target domain
    M : array-like, shape (ns, nt)
        loss matrix
    reg : float
        Regularization term >0
    G0 :  array-like, shape (ns,nt), optional
        initial guess (default is indep joint density)
    line_search: function,
        Function to find the optimal step.
        Default is None and calls a wrapper to line_search_armijo.
    numItermax : int, optional
        Max number of iterations
    stopThr : float, optional
        Stop threshold on the relative variation (>0)