Function wda

ot/dr.py:141–283

r""" Wasserstein Discriminant Analysis :ref:`[11] ` The function solves the following optimization problem: .. math:: \mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad \frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \m

(
    X,
    y,
    p=2,
    reg=1,
    k=10,
    solver=None,
    sinkhorn_method="sinkhorn",
    maxiter=100,
    verbose=0,
    P0=None,
    normalize=False,
)

def wda(
    X,
    y,
    p=2,
    reg=1,
    k=10,
    solver=None,
    sinkhorn_method="sinkhorn",
    maxiter=100,
    verbose=0,
    P0=None,
    normalize=False,
):
    r"""
    Wasserstein Discriminant Analysis :ref:`[11] <references-wda>`

    The function solves the following optimization problem:

    .. math::
        \mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad
        \frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}

    where :

    - :math:`P` is a linear projection operator in the Stiefel(`p`, `d`) manifold
    - :math:`W` is entropic regularized Wasserstein distances
    - :math:`\mathbf{X}^i` are samples in the dataset corresponding to class i

    **Choosing a Sinkhorn solver**

    By default and when using a regularization parameter that is not too small
    the default sinkhorn solver should be enough. If you need to use a small
    regularization to get sparse cost matrices, you should use the
    :py:func:`ot.dr.sinkhorn_log` solver that will avoid numerical
    errors, but can be slow in practice.

    Parameters
    ----------
    X : ndarray, shape (n, d)
        Training samples.
    y : ndarray, shape (n,)
        Labels for training samples.
    p : int, optional
        Size of dimensionality reduction.
    reg : float, optional
        Regularization term >0 (entropic regularization)
    solver : None | str, optional
        None for steepest descent or 'TrustRegions' for trust regions algorithm
        else should be a pymanopt.solvers
    sinkhorn_method : str
        method used for the Sinkhorn solver, either 'sinkhorn' or 'sinkhorn_log'
    P0 : ndarray, shape (d, p)
        Initial starting point for projection.
    normalize : bool, optional
        Normalize the Wasserstaiun distance by the average distance on P0 (default : False)
    verbose : int, optional
        Print information along iterations.