Function generic_conditional_gradient

ot/optim.py:143–409

r""" Solve the general regularized OT problem or its semi-relaxed version with conditional gradient or generalized conditional gradient depending on the provided linear program solver. The function solves the following optimization problem if set as a conditional gradient:

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

def generic_conditional_gradient(
    a,
    b,
    M,
    f,
    df,
    reg1,
    reg2,
    lp_solver,
    line_search,
    G0=None,
    numItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    verbose=False,
    log=False,
    nx=None,
    **kwargs,
):
    r"""
    Solve the general regularized OT problem or its semi-relaxed version with
    conditional gradient or generalized conditional gradient depending on the
    provided linear program solver.

        The function solves the following optimization problem if set as a conditional gradient:

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

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

             \gamma^T \mathbf{1} &= \mathbf{b} (optional constraint)

             \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>`

        The function solves the following optimization problem if set a generalized conditional gradient:

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

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

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

             \gamma &\geq 0

    where :