Function regularization_path

ot/regpath.py:836–912

r"""This function provides all the solutions of the regularization path \ of the l2-UOT problem :ref:`[41] `. The problem to optimize is the Lasso reformulation of the l2-penalized UOT: .. math:: \min_t \gamma \mathbf{c}^T \mathbf{t} + 0.5 *

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

def regularization_path(
    a: np.array, b: np.array, C: np.array, reg=1e-4, semi_relaxed=False, itmax=50000
):
    r"""This function provides all the solutions of the regularization path \
    of the l2-UOT problem :ref:`[41] <references-regpath>`.

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

    .. math::
        \min_t \gamma \mathbf{c}^T \mathbf{t}
                + 0.5 * \|{H} \mathbf{t} - \mathbf{y}\|_2^2

        s.t.
            \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 coefficient
    - :math:`\mathbf{y}` is the concatenation of vectors :math:`\mathbf{a}` \
      and :math:`\mathbf{b}`, defined as \
      :math:`\mathbf{y}^T = [\mathbf{a}^T \mathbf{b}^T]`
    - :math:`{H}` is a design matrix, see :ref:`[41] <references-regpath>` \
      for the design of :math:`{H}`. The matrix product :math:`H\mathbf{t}` \
      computes both the source marginal and the target marginals.
    - :math:`\mathbf{t}` is the flattened version of the transport matrix

    For the semi-relaxed problem, it optimizes the Lasso reformulation of the
    l2-penalized UOT:

    .. math::

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

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

            \mathbf{t} \geq 0


    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
    semi_relaxed : bool (optional)
        Give the semi-relaxed path if True
    itmax: int (optional)
        Maximum number of iteration

    Returns