Function joint_OT_mapping_kernel

ot/mapping.py:646–894

r"""Joint OT and nonlinear mapping estimation with kernels as proposed in :ref:`[8] `. The function solves the following optimization problem: .. math:: \min_{\gamma, L\in\mathcal{H}}\quad \|L(\mathbf{X_s}) - n_s\gamma \mathbf{X_t}\|^

(
    xs,
    xt,
    mu=1,
    eta=0.001,
    kerneltype="gaussian",
    sigma=1,
    bias=False,
    verbose=False,
    verbose2=False,
    numItermax=100,
    numInnerItermax=10,
    stopInnerThr=1e-6,
    stopThr=1e-5,
    log=False,
    **kwargs,
)

def joint_OT_mapping_kernel(
    xs,
    xt,
    mu=1,
    eta=0.001,
    kerneltype="gaussian",
    sigma=1,
    bias=False,
    verbose=False,
    verbose2=False,
    numItermax=100,
    numInnerItermax=10,
    stopInnerThr=1e-6,
    stopThr=1e-5,
    log=False,
    **kwargs,
):
    r"""Joint OT and nonlinear mapping estimation with kernels as proposed in
    :ref:`[8] <references-joint-OT-mapping-kernel>`.

    The function solves the following optimization problem:

    .. math::
        \min_{\gamma, L\in\mathcal{H}}\quad \|L(\mathbf{X_s}) -
        n_s\gamma \mathbf{X_t}\|^2_F + \mu \langle \gamma, \mathbf{M} \rangle_F +
        \eta \|L\|^2_\mathcal{H}

        s.t. \ \gamma \mathbf{1} = \mathbf{a}

             \gamma^T \mathbf{1} = \mathbf{b}

             \gamma \geq 0


    where :

    - :math:`\mathbf{M}` is the (`ns`, `nt`) squared euclidean cost matrix between samples in
      :math:`\mathbf{X_s}` and :math:`\mathbf{X_t}` (scaled by :math:`n_s`)
    - :math:`L` is a :math:`n_s \times d` linear operator on a kernel matrix that
      approximates the barycentric mapping
    - :math:`\mathbf{a}` and :math:`\mathbf{b}` are uniform source and target weights

    The problem consist in solving jointly an optimal transport matrix
    :math:`\gamma` and the nonlinear mapping that fits the barycentric mapping
    :math:`n_s\gamma \mathbf{X_t}`.

    One can also estimate a mapping with constant bias (see supplementary
    material of :ref:`[8] <references-joint-OT-mapping-kernel>`) using the bias optional argument.

    The algorithm used for solving the problem is the block coordinate
    descent that alternates between updates of :math:`\mathbf{G}` (using conditional gradient)
    and the update of :math:`\mathbf{L}` using a classical kernel least square solver.


    Parameters
    ----------
    xs : array-like (ns,d)
        samples in the source domain