content: """Module :mod:`sklearn.kernel_ridge` implements kernel ridge regression."""
  
  # Authors: Mathieu Blondel <mathieu@mblondel.org>
  #          Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
  # License: BSD 3 clause
  
  import numpy as np
  
  from .base import BaseEstimator, RegressorMixin
  from .metrics.pairwise import pairwise_kernels
  from .linear_model.ridge import _solve_cholesky_kernel
  from .utils import check_array, check_X_y
  from .utils.validation import check_is_fitted
  
  
  class KernelRidge(BaseEstimator, RegressorMixin):
      """Kernel ridge regression.
  
      Kernel ridge regression (KRR) combines ridge regression (linear least
      squares with l2-norm regularization) with the kernel trick. It thus
      learns a linear function in the space induced by the respective kernel and
      the data. For non-linear kernels, this corresponds to a non-linear
      function in the original space.
  
      The form of the model learned by KRR is identical to support vector
      regression (SVR). However, different loss functions are used: KRR uses
      squared error loss while support vector regression uses epsilon-insensitive
      loss, both combined with l2 regularization. In contrast to SVR, fitting a
      KRR model can be done in closed-form and is typically faster for
      medium-sized datasets. On the other  hand, the learned model is non-sparse
      and thus slower than SVR, which learns a sparse model for epsilon > 0, at
      prediction-time.
  
      This estimator has built-in support for multi-variate regression
      (i.e., when y is a 2d-array of shape [n_samples, n_targets]).
  
      Read more in the :ref:`User Guide <kernel_ridge>`.
  
      Parameters
      ----------
      alpha : {float, array-like}, shape = [n_targets]
          Small positive values of alpha improve the conditioning of the problem
          and reduce the variance of the estimates.  Alpha corresponds to
          ``(2*C)^-1`` in other linear models such as LogisticRegression or
          LinearSVC. If an array is passed, penalties are assumed to be specific
          to the targets. Hence they must correspond in number.
  
      kernel : string or callable, default="linear"
          Kernel mapping used internally. A callable should accept two arguments
          and the keyword arguments passed to this object as kernel_params, and
          should return a floating point number.
  
      gamma : float, default=None
          Gamma parameter for the RBF, laplacian, polynomial, exponential chi2
          and sigmoid kernels. Interpretation of the default value is left to
          the kernel; see the documentation for sklearn.metrics.pairwise.
          Ignored by other kernels.
  
      degree : float, default=3
          Degree of the polynomial kernel. Ignored by other kernels.
  
      coef0 : float, default=1
          Zero coefficient for polynomial and sigmoid kernels.
          Ignored by other kernels.
  
      kernel_params : mapping of string to any, optional
          Additional parameters (keyword arguments) for kernel function passed
          as callable object.
  
      Attributes
      ----------
      dual_coef_ : array, shape = [n_samples] or [n_samples, n_targets]
          Representation of weight vector(s) in kernel space
  
      X_fit_ : {array-like, sparse matrix}, shape = [n_samples, n_features]
          Training data, which is also required for prediction
  
      References
      ----------
      * Kevin P. Murphy
        "Machine Learning: A Probabilistic Perspective", The MIT Press
        chapter 14.4.3, pp. 492-493
  
      See also
      --------
      sklearn.linear_model.Ridge:
          Linear ridge regression.
      sklearn.svm.SVR:
          Support Vector Regression implemented using libsvm.
  
      Examples
      --------
      >>> from sklearn.kernel_ridge import KernelRidge
      >>> import numpy as np
      >>> n_samples, n_features = 10, 5
      >>> rng = np.random.RandomState(0)
      >>> y = rng.randn(n_samples)
      >>> X = rng.randn(n_samples, n_features)
      >>> clf = KernelRidge(alpha=1.0)
      >>> clf.fit(X, y) # doctest: +NORMALIZE_WHITESPACE
      KernelRidge(alpha=1.0, coef0=1, degree=3, gamma=None, kernel='linear',
                  kernel_params=None)
      """
      def __init__(self, alpha=1, kernel="linear", gamma=None, degree=3, coef0=1,
                   kernel_params=None):
          self.alpha = alpha
          self.kernel = kernel
          self.gamma = gamma
          self.degree = degree
          self.coef0 = coef0
          self.kernel_params = kernel_params
  
      def _get_kernel(self, X, Y=None):
          if callable(self.kernel):
              params = self.kernel_params or {}
          else:
              params = {"gamma": self.gamma,
                        "degree": self.degree,
                        "coef0": self.coef0}
          return pairwise_kernels(X, Y, metric=self.kernel,
                                  filter_params=True, **params)
  
      @property
      def _pairwise(self):
          return self.kernel == "precomputed"
  
      def fit(self, X, y=None, sample_weight=None):
          """Fit Kernel Ridge regression model
  
          Parameters
          ----------
          X : {array-like, sparse matrix}, shape = [n_samples, n_features]
              Training data
  
          y : array-like, shape = [n_samples] or [n_samples, n_targets]
              Target values
  
          sample_weight : float or array-like of shape [n_samples]
              Individual weights for each sample, ignored if None is passed.
  
          Returns
          -------
          self : returns an instance of self.
          """
          # Convert data
          X, y = check_X_y(X, y, accept_sparse=("csr", "csc"), multi_output=True,
                           y_numeric=True)
          if sample_weight is not None and not isinstance(sample_weight, float):
              sample_weight = check_array(sample_weight, ensure_2d=False)
  
          K = self._get_kernel(X)
          alpha = np.atleast_1d(self.alpha)
  
          ravel = False
          if len(y.shape) == 1:
              y = y.reshape(-1, 1)
              ravel = True
  
          copy = self.kernel == "precomputed"
          self.dual_coef_ = _solve_cholesky_kernel(K, y, alpha,
                                                   sample_weight,
                                                   copy)
          if ravel:
              self.dual_coef_ = self.dual_coef_.ravel()
  
          self.X_fit_ = X
  
          return self
  
      def predict(self, X):
          """Predict using the kernel ridge model
  
          Parameters
          ----------
          X : {array-like, sparse matrix}, shape = [n_samples, n_features]
              Samples.
  
          Returns
          -------
          C : array, shape = [n_samples] or [n_samples, n_targets]
              Returns predicted values.
          """
          check_is_fitted(self, ["X_fit_", "dual_coef_"])
          K = self._get_kernel(X, self.X_fit_)
          return np.dot(K, self.dual_coef_)
  