Function estimate_dual_null_weights

ot/lp/_network_simplex.py:80–163

r"""Estimate feasible values for 0-weighted dual potentials The feasible values are computed efficiently but rather coarsely. .. warning:: This function is necessary because the C++ solver in `emd_c` discards all samples in the distributions with zeros weights. This

(alpha0, beta0, a, b, M)

def estimate_dual_null_weights(alpha0, beta0, a, b, M):
    r"""Estimate feasible values for 0-weighted dual potentials

    The feasible values are computed efficiently but rather coarsely.

    .. warning::
        This function is necessary because the C++ solver in `emd_c`
        discards all samples in the distributions with
        zeros weights. This means that while the primal variable (transport
        matrix) is exact, the solver only returns feasible dual potentials
        on the samples with weights different from zero.

    First we compute the constraints violations:

    .. math::
        \mathbf{V} = \alpha + \beta^T - \mathbf{M}

    Next we compute the max amount of violation per row (:math:`\alpha`) and
    columns (:math:`beta`)

    .. math::
        \mathbf{v^a}_i = \max_j \mathbf{V}_{i,j}

        \mathbf{v^b}_j = \max_i \mathbf{V}_{i,j}

    Finally we update the dual potential with 0 weights if a
    constraint is violated

    .. math::
        \alpha_i = \alpha_i - \mathbf{v^a}_i \quad \text{ if } \mathbf{a}_i=0 \text{ and } \mathbf{v^a}_i>0

        \beta_j = \beta_j - \mathbf{v^b}_j \quad \text{ if } \mathbf{b}_j=0 \text{ and } \mathbf{v^b}_j > 0

    In the end the dual potentials are centered using function
    :py:func:`ot.lp.center_ot_dual`.

    Note that all those updates do not change the objective value of the
    solution but provide dual potentials that do not violate the constraints.

    Parameters
    ----------
    alpha0 : (ns,) numpy.ndarray, float64
        Source dual potential
    beta0 : (nt,) numpy.ndarray, float64
        Target dual potential
    alpha0 : (ns,) numpy.ndarray, float64
        Source dual potential
    beta0 : (nt,) numpy.ndarray, float64
        Target dual potential
    a : (ns,) numpy.ndarray, float64
        Source distribution (uniform weights if empty list)
    b : (nt,) numpy.ndarray, float64
        Target distribution (uniform weights if empty list)
    M : (ns,nt) numpy.ndarray, float64
        Loss matrix (c-order array with type float64)

    Returns
    -------