Function nearest_brenier_potential_predict_bounds

ot/mapping.py:228–421

r""" Compute the values of the lower and upper bounding potentials at the input points Y, using the potential optimal values phi at X and their gradients G at X. The 'lower' potential corresponds to the method from :ref:`[58]`, Equation 2, while the bounding property and 'upper' potentia

(
    X,
    phi,
    G,
    Y,
    X_classes=None,
    Y_classes=None,
    strongly_convex_constant=0.6,
    gradient_lipschitz_constant=1.4,
    log=False,
    solver=None,
)

def nearest_brenier_potential_predict_bounds(
    X,
    phi,
    G,
    Y,
    X_classes=None,
    Y_classes=None,
    strongly_convex_constant=0.6,
    gradient_lipschitz_constant=1.4,
    log=False,
    solver=None,
):
    r"""
    Compute the values of the lower and upper bounding potentials at the input points Y, using the potential optimal
    values phi at X and their gradients G at X. The 'lower' potential corresponds to the method from :ref:`[58]`,
    Equation 2, while the bounding property and 'upper' potential come from :ref:`[59]`, Theorem 3.14 (taking into
    account the fact that this theorem's statement has a min instead of a max, which is a typo). Both potentials are
    optimal for the SSNB problem.

    If :math:`I_k` is the subset of :math:`[n]` of the i such that :math:`x_i` is in the partition (or class)
    :math:`E_k`, for each :math:`y \in E_k`, this function solves the convex QCQP problems,
    respectively for l: 'lower' and u: 'upper':

    .. math::
        :nowrap:

        \begin{gather*}
        (\varphi_{l}(x), \nabla \varphi_l(x)) = \text{argmin}\ t, \\
        t\in \mathbb{R},\; g\in \mathbb{R}^d, \\
        \text{s.t.} \forall j \in I_k,\; t-\varphi_j - \langle g_j, y-x_j \rangle \geq c_1\|g - g_j\|_2^2
        + c_2\|y-x_j\|_2^2 - c_3\langle g_j-g, x_j -y \rangle.
        \end{gather*}

    .. math::
        :nowrap:

        \begin{gather*}
        (\varphi_{u}(x), \nabla \varphi_u(x)) = \text{argmax}\ t, \\
        t\in \mathbb{R},\; g\in \mathbb{R}^d, \\
        \text{s.t.} \forall i \in I_k,\; \varphi_i^* -t - \langle g, x_i-y \rangle \geq c_1\|g_i - g\|_2^2
        + c_2\|x_i-y\|_2^2 - c_3\langle g-g_i, y -x_i \rangle.
        \end{gather*}

    The constants :math:`c_1, c_2, c_3` only depend on `strongly_convex_constant` and `gradient_lipschitz_constant`.

    .. warning:: This function requires the CVXPY library
    .. warning:: Accepts any backend but will convert to Numpy then back to the backend.

    Parameters
    ----------
    X : array-like (n, d)
        reference points used to compute the optimal values phi and G
    X_classes : array-like (n,)
        classes of the reference points
    phi : array-like (n,)
        optimal values of the potential at the points X
    G : array-like (n, d)
        optimal values of the gradients at the points X