Function semi_relaxed_path

ot/regpath.py:681–833

r"""This function gives the regularization path of semi-relaxed l2-UOT problem. The problem to optimize is the Lasso reformulation of the l2-penalized UOT: .. math:: \min_t \gamma \mathbf{c}^T t + 0.5 * \|H_r \mathbf{t} - \mathbf{a}\|_2^2 s.t.

(a: np.array, b: np.array, C: np.array, reg=1e-4, itmax=50000)

def semi_relaxed_path(a: np.array, b: np.array, C: np.array, reg=1e-4, itmax=50000):
    r"""This function gives the regularization path of semi-relaxed
    l2-UOT problem.

    The problem to optimize is the Lasso reformulation of the l2-penalized UOT:

    .. math::

        \min_t \gamma \mathbf{c}^T t
               + 0.5 * \|H_r \mathbf{t} - \mathbf{a}\|_2^2

        s.t.
            H_c \mathbf{t} = \mathbf{b}

            \mathbf{t} \geq 0

    where :

    - :math:`\mathbf{c}` is the flattened version of the cost matrix \
      :math:`C`
    - :math:`\gamma = 1/\lambda` is the l2-regularization parameter
    - :math:`H_r` is  a  matrix that computes the sum along the rows of \
      the transport plan :math:`T`
    - :math:`H_c` is  a  matrix that computes the sum along the columns of \
      the transport plan :math:`T`
    - :math:`\mathbf{t}` is the flattened version of the transport plan \
      :math:`T`

    Parameters
    ----------
    a : np.ndarray (dim_a,)
        Histogram of dimension dim_a
    b : np.ndarray (dim_b,)
        Histogram of dimension dim_b
    C : np.ndarray, shape (dim_a, dim_b)
        Cost matrix
    reg: float (optional)
        l2-regularization coefficient
    itmax: int (optional)
        Maximum number of iteration

    Returns
    -------
    t : np.ndarray (dim_a*dim_b, )
        Flattened vector of the (unregularized) optimal transport matrix
    t_list : list
        List of all the optimal transport vectors of the regularization path
    gamma_list : list
        List of the regularization parameters in the path

    Examples
    --------
    >>> import ot
    >>> import numpy as np
    >>> n = 3
    >>> xs = np.array([1., 2., 3.]).reshape((n, 1))
    >>> xt = np.array([5., 6., 7.]).reshape((n, 1))
    >>> C = ot.dist(xs, xt)