Function partial_cg

ot/optim.py:669–805

r""" Solve the general regularized partial 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.

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

def partial_cg(
    a,
    b,
    a_extended,
    b_extended,
    M,
    reg,
    f,
    df,
    G0=None,
    line_search=line_search_armijo,
    numItermax=200,
    stopThr=1e-9,
    stopThr2=1e-9,
    warn=True,
    verbose=False,
    log=False,
    **kwargs,
):
    r"""
    Solve the general regularized partial 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 \mathbf{1} &= \mathbf{b}

             \mathbf{1}^T \gamma^T \mathbf{1} = m &\leq \min\{\|\mathbf{p}\|_1, \|\mathbf{q}\|_1\}

             \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
    - `m` is the amount of mass to be transported

    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
    a_extended : array-like, shape (ns + nb_dummies,)
        samples weights in the source domain with added dummy nodes
    b_extended : array-like, shape (nt + nb_dummies,)
        currently estimated samples weights in the target domain with added dummy nodes
    M : array-like, shape (ns, nt)
        loss matrix
    reg : float