content: import numpy as np
  import scipy.sparse as sp
  
  from sklearn.datasets import make_regression
  from sklearn.linear_model import Ridge
  from sklearn.kernel_ridge import KernelRidge
  from sklearn.metrics.pairwise import pairwise_kernels
  from sklearn.utils.testing import ignore_warnings
  
  from sklearn.utils.testing import assert_array_almost_equal
  
  
  X, y = make_regression(n_features=10)
  Xcsr = sp.csr_matrix(X)
  Xcsc = sp.csc_matrix(X)
  Y = np.array([y, y]).T
  
  
  def test_kernel_ridge():
      pred = Ridge(alpha=1, fit_intercept=False).fit(X, y).predict(X)
      pred2 = KernelRidge(kernel="linear", alpha=1).fit(X, y).predict(X)
      assert_array_almost_equal(pred, pred2)
  
  
  def test_kernel_ridge_csr():
      pred = Ridge(alpha=1, fit_intercept=False,
                   solver="cholesky").fit(Xcsr, y).predict(Xcsr)
      pred2 = KernelRidge(kernel="linear", alpha=1).fit(Xcsr, y).predict(Xcsr)
      assert_array_almost_equal(pred, pred2)
  
  
  def test_kernel_ridge_csc():
      pred = Ridge(alpha=1, fit_intercept=False,
                   solver="cholesky").fit(Xcsc, y).predict(Xcsc)
      pred2 = KernelRidge(kernel="linear", alpha=1).fit(Xcsc, y).predict(Xcsc)
      assert_array_almost_equal(pred, pred2)
  
  
  def test_kernel_ridge_singular_kernel():
      # alpha=0 causes a LinAlgError in computing the dual coefficients,
      # which causes a fallback to a lstsq solver. This is tested here.
      pred = Ridge(alpha=0, fit_intercept=False).fit(X, y).predict(X)
      kr = KernelRidge(kernel="linear", alpha=0)
      ignore_warnings(kr.fit)(X, y)
      pred2 = kr.predict(X)
      assert_array_almost_equal(pred, pred2)
  
  
  def test_kernel_ridge_precomputed():
      for kernel in ["linear", "rbf", "poly", "cosine"]:
          K = pairwise_kernels(X, X, metric=kernel)
          pred = KernelRidge(kernel=kernel).fit(X, y).predict(X)
          pred2 = KernelRidge(kernel="precomputed").fit(K, y).predict(K)
          assert_array_almost_equal(pred, pred2)
  
  
  def test_kernel_ridge_precomputed_kernel_unchanged():
      K = np.dot(X, X.T)
      K2 = K.copy()
      KernelRidge(kernel="precomputed").fit(K, y)
      assert_array_almost_equal(K, K2)
  
  
  def test_kernel_ridge_sample_weights():
      K = np.dot(X, X.T)  # precomputed kernel
      sw = np.random.RandomState(0).rand(X.shape[0])
  
      pred = Ridge(alpha=1,
                   fit_intercept=False).fit(X, y, sample_weight=sw).predict(X)
      pred2 = KernelRidge(kernel="linear",
                          alpha=1).fit(X, y, sample_weight=sw).predict(X)
      pred3 = KernelRidge(kernel="precomputed",
                          alpha=1).fit(K, y, sample_weight=sw).predict(K)
      assert_array_almost_equal(pred, pred2)
      assert_array_almost_equal(pred, pred3)
  
  
  def test_kernel_ridge_multi_output():
      pred = Ridge(alpha=1, fit_intercept=False).fit(X, Y).predict(X)
      pred2 = KernelRidge(kernel="linear", alpha=1).fit(X, Y).predict(X)
      assert_array_almost_equal(pred, pred2)
  
      pred3 = KernelRidge(kernel="linear", alpha=1).fit(X, y).predict(X)
      pred3 = np.array([pred3, pred3]).T
      assert_array_almost_equal(pred2, pred3)
  