Function c_transform_entropic

ot/stochastic.py:225–278

r""" The goal is to recover u from the c-transform. The function computes the c-transform of a dual variable from the other dual variable: .. math:: \mathbf{u} = \mathbf{v}^{c,reg} = - \mathrm{reg} \sum_j \mathbf{b}_j \exp\left( \frac{\mathbf{v} - \mathbf{M}}{\mathr

(b, M, reg, beta)

def c_transform_entropic(b, M, reg, beta):
    r"""
    The goal is to recover u from the c-transform.

    The function computes the c-transform of a dual variable from the other
    dual variable:

    .. math::
        \mathbf{u} = \mathbf{v}^{c,reg} = - \mathrm{reg} \sum_j \mathbf{b}_j
        \exp\left( \frac{\mathbf{v} - \mathbf{M}}{\mathrm{reg}} \right)

    Where :

    - :math:`\mathbf{M}` is the (`ns`, `nt`) metric cost matrix
    - :math:`\mathbf{u}`, :math:`\mathbf{v}` are dual variables in :math:`\mathbb{R}^{ns} \times \mathbb{R}^{nt}`
    - reg is the regularization term

    It is used to recover an optimal u from optimal v solving the semi dual
    problem, see Proposition 2.1 of :ref:`[18] <references-c-transform-entropic>`


    Parameters
    ----------
    b : ndarray, shape (nt,)
        Target measure
    M : ndarray, shape (ns, nt)
        Cost matrix
    reg : float
        Regularization term > 0
    v : ndarray, shape (nt,)
        Dual variable.

    Returns
    -------
    u : ndarray, shape (`ns`,)
        Dual variable.


    .. _references-c-transform-entropic:
    References
    ----------
    .. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016)
        Stochastic Optimization for Large-scale Optimal Transport.
        Advances in Neural Information Processing Systems (2016).
    """

    n_source = np.shape(M)[0]
    alpha = np.zeros(n_source)
    for i in range(n_source):
        r = M[i, :] - beta
        min_r = np.min(r)
        exp_beta = np.exp(-(r - min_r) / reg) * b
        alpha[i] = min_r - reg * np.log(np.sum(exp_beta))
    return alpha


def solve_semi_dual_entropic(