code: import sys
  from io import StringIO
  import numpy as np
  from numpy.testing import assert_allclose
  import scipy.sparse as sp
  import pytest
  
  from sklearn.neighbors import NearestNeighbors
  from sklearn.neighbors import kneighbors_graph
  from sklearn.exceptions import EfficiencyWarning
  from sklearn.utils._testing import ignore_warnings
  from sklearn.utils._testing import assert_almost_equal
  from sklearn.utils._testing import assert_array_equal
  from sklearn.utils._testing import assert_array_almost_equal
  from sklearn.utils._testing import skip_if_32bit
  from sklearn.utils import check_random_state
  from sklearn.manifold._t_sne import _joint_probabilities
  from sklearn.manifold._t_sne import _joint_probabilities_nn
  from sklearn.manifold._t_sne import _kl_divergence
  from sklearn.manifold._t_sne import _kl_divergence_bh
  from sklearn.manifold._t_sne import _gradient_descent
  from sklearn.manifold._t_sne import trustworthiness
  from sklearn.manifold import TSNE
  
  # mypy error: Module 'sklearn.manifold' has no attribute '_barnes_hut_tsne'
  from sklearn.manifold import _barnes_hut_tsne  # type: ignore
  from sklearn.manifold._utils import _binary_search_perplexity
  from sklearn.datasets import make_blobs
  from scipy.optimize import check_grad
  from scipy.spatial.distance import pdist
  from scipy.spatial.distance import squareform
  from sklearn.metrics.pairwise import pairwise_distances
  from sklearn.metrics.pairwise import manhattan_distances
  from sklearn.metrics.pairwise import cosine_distances
  
  
  x = np.linspace(0, 1, 10)
  xx, yy = np.meshgrid(x, x)
  X_2d_grid = np.hstack(
      [
          xx.ravel().reshape(-1, 1),
          yy.ravel().reshape(-1, 1),
      ]
  )
  
  pytestmark = pytest.mark.filterwarnings(
      "ignore:The PCA initialization in TSNE will change to have the standard deviation",
  )
  
  
  def test_gradient_descent_stops():
      # Test stopping conditions of gradient descent.
      class ObjectiveSmallGradient:
          def __init__(self):
              self.it = -1
  
          def __call__(self, _, compute_error=True):
              self.it += 1
              return (10 - self.it) / 10.0, np.array([1e-5])
  
      def flat_function(_, compute_error=True):
          return 0.0, np.ones(1)
  
      # Gradient norm
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          _, error, it = _gradient_descent(
              ObjectiveSmallGradient(),
              np.zeros(1),
              0,
              n_iter=100,
              n_iter_without_progress=100,
              momentum=0.0,
              learning_rate=0.0,
              min_gain=0.0,
              min_grad_norm=1e-5,
              verbose=2,
          )
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
      assert error == 1.0
      assert it == 0
      assert "gradient norm" in out
  
      # Maximum number of iterations without improvement
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          _, error, it = _gradient_descent(
              flat_function,
              np.zeros(1),
              0,
              n_iter=100,
              n_iter_without_progress=10,
              momentum=0.0,
              learning_rate=0.0,
              min_gain=0.0,
              min_grad_norm=0.0,
              verbose=2,
          )
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
      assert error == 0.0
      assert it == 11
      assert "did not make any progress" in out
  
      # Maximum number of iterations
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          _, error, it = _gradient_descent(
              ObjectiveSmallGradient(),
              np.zeros(1),
              0,
              n_iter=11,
              n_iter_without_progress=100,
              momentum=0.0,
              learning_rate=0.0,
              min_gain=0.0,
              min_grad_norm=0.0,
              verbose=2,
          )
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
      assert error == 0.0
      assert it == 10
      assert "Iteration 10" in out
  
  
  def test_binary_search():
      # Test if the binary search finds Gaussians with desired perplexity.
      random_state = check_random_state(0)
      data = random_state.randn(50, 5)
      distances = pairwise_distances(data).astype(np.float32)
      desired_perplexity = 25.0
      P = _binary_search_perplexity(distances, desired_perplexity, verbose=0)
      P = np.maximum(P, np.finfo(np.double).eps)
      mean_perplexity = np.mean(
          [np.exp(-np.sum(P[i] * np.log(P[i]))) for i in range(P.shape[0])]
      )
      assert_almost_equal(mean_perplexity, desired_perplexity, decimal=3)
  
  
  def test_binary_search_underflow():
      # Test if the binary search finds Gaussians with desired perplexity.
      # A more challenging case than the one above, producing numeric
      # underflow in float precision (see issue #19471 and PR #19472).
      random_state = check_random_state(42)
      data = random_state.randn(1, 90).astype(np.float32) + 100
      desired_perplexity = 30.0
      P = _binary_search_perplexity(data, desired_perplexity, verbose=0)
      perplexity = 2 ** -np.nansum(P[0, 1:] * np.log2(P[0, 1:]))
      assert_almost_equal(perplexity, desired_perplexity, decimal=3)
  
  
  def test_binary_search_neighbors():
      # Binary perplexity search approximation.
      # Should be approximately equal to the slow method when we use
      # all points as neighbors.
      n_samples = 200
      desired_perplexity = 25.0
      random_state = check_random_state(0)
      data = random_state.randn(n_samples, 2).astype(np.float32, copy=False)
      distances = pairwise_distances(data)
      P1 = _binary_search_perplexity(distances, desired_perplexity, verbose=0)
  
      # Test that when we use all the neighbors the results are identical
      n_neighbors = n_samples - 1
      nn = NearestNeighbors().fit(data)
      distance_graph = nn.kneighbors_graph(n_neighbors=n_neighbors, mode="distance")
      distances_nn = distance_graph.data.astype(np.float32, copy=False)
      distances_nn = distances_nn.reshape(n_samples, n_neighbors)
      P2 = _binary_search_perplexity(distances_nn, desired_perplexity, verbose=0)
  
      indptr = distance_graph.indptr
      P1_nn = np.array(
          [
              P1[k, distance_graph.indices[indptr[k] : indptr[k + 1]]]
              for k in range(n_samples)
          ]
      )
      assert_array_almost_equal(P1_nn, P2, decimal=4)
  
      # Test that the highest P_ij are the same when fewer neighbors are used
      for k in np.linspace(150, n_samples - 1, 5):
          k = int(k)
          topn = k * 10  # check the top 10 * k entries out of k * k entries
          distance_graph = nn.kneighbors_graph(n_neighbors=k, mode="distance")
          distances_nn = distance_graph.data.astype(np.float32, copy=False)
          distances_nn = distances_nn.reshape(n_samples, k)
          P2k = _binary_search_perplexity(distances_nn, desired_perplexity, verbose=0)
          assert_array_almost_equal(P1_nn, P2, decimal=2)
          idx = np.argsort(P1.ravel())[::-1]
          P1top = P1.ravel()[idx][:topn]
          idx = np.argsort(P2k.ravel())[::-1]
          P2top = P2k.ravel()[idx][:topn]
          assert_array_almost_equal(P1top, P2top, decimal=2)
  
  
  def test_binary_perplexity_stability():
      # Binary perplexity search should be stable.
      # The binary_search_perplexity had a bug wherein the P array
      # was uninitialized, leading to sporadically failing tests.
      n_neighbors = 10
      n_samples = 100
      random_state = check_random_state(0)
      data = random_state.randn(n_samples, 5)
      nn = NearestNeighbors().fit(data)
      distance_graph = nn.kneighbors_graph(n_neighbors=n_neighbors, mode="distance")
      distances = distance_graph.data.astype(np.float32, copy=False)
      distances = distances.reshape(n_samples, n_neighbors)
      last_P = None
      desired_perplexity = 3
      for _ in range(100):
          P = _binary_search_perplexity(distances.copy(), desired_perplexity, verbose=0)
          P1 = _joint_probabilities_nn(distance_graph, desired_perplexity, verbose=0)
          # Convert the sparse matrix to a dense one for testing
          P1 = P1.toarray()
          if last_P is None:
              last_P = P
              last_P1 = P1
          else:
              assert_array_almost_equal(P, last_P, decimal=4)
              assert_array_almost_equal(P1, last_P1, decimal=4)
  
  
  def test_gradient():
      # Test gradient of Kullback-Leibler divergence.
      random_state = check_random_state(0)
  
      n_samples = 50
      n_features = 2
      n_components = 2
      alpha = 1.0
  
      distances = random_state.randn(n_samples, n_features).astype(np.float32)
      distances = np.abs(distances.dot(distances.T))
      np.fill_diagonal(distances, 0.0)
      X_embedded = random_state.randn(n_samples, n_components).astype(np.float32)
  
      P = _joint_probabilities(distances, desired_perplexity=25.0, verbose=0)
  
      def fun(params):
          return _kl_divergence(params, P, alpha, n_samples, n_components)[0]
  
      def grad(params):
          return _kl_divergence(params, P, alpha, n_samples, n_components)[1]
  
      assert_almost_equal(check_grad(fun, grad, X_embedded.ravel()), 0.0, decimal=5)
  
  
  def test_trustworthiness():
      # Test trustworthiness score.
      random_state = check_random_state(0)
  
      # Affine transformation
      X = random_state.randn(100, 2)
      assert trustworthiness(X, 5.0 + X / 10.0) == 1.0
  
      # Randomly shuffled
      X = np.arange(100).reshape(-1, 1)
      X_embedded = X.copy()
      random_state.shuffle(X_embedded)
      assert trustworthiness(X, X_embedded) < 0.6
  
      # Completely different
      X = np.arange(5).reshape(-1, 1)
      X_embedded = np.array([[0], [2], [4], [1], [3]])
      assert_almost_equal(trustworthiness(X, X_embedded, n_neighbors=1), 0.2)
  
  
  @pytest.mark.parametrize("method", ["exact", "barnes_hut"])
  @pytest.mark.parametrize("init", ("random", "pca"))
  def test_preserve_trustworthiness_approximately(method, init):
      # Nearest neighbors should be preserved approximately.
      random_state = check_random_state(0)
      n_components = 2
      X = random_state.randn(50, n_components).astype(np.float32)
      tsne = TSNE(
          n_components=n_components,
          init=init,
          random_state=0,
          method=method,
          n_iter=700,
          learning_rate="auto",
      )
      X_embedded = tsne.fit_transform(X)
      t = trustworthiness(X, X_embedded, n_neighbors=1)
      assert t > 0.85
  
  
  def test_optimization_minimizes_kl_divergence():
      """t-SNE should give a lower KL divergence with more iterations."""
      random_state = check_random_state(0)
      X, _ = make_blobs(n_features=3, random_state=random_state)
      kl_divergences = []
      for n_iter in [250, 300, 350]:
          tsne = TSNE(
              n_components=2,
              init="random",
              perplexity=10,
              learning_rate=100.0,
              n_iter=n_iter,
              random_state=0,
          )
          tsne.fit_transform(X)
          kl_divergences.append(tsne.kl_divergence_)
      assert kl_divergences[1] <= kl_divergences[0]
      assert kl_divergences[2] <= kl_divergences[1]
  
  
  @pytest.mark.parametrize("method", ["exact", "barnes_hut"])
  def test_fit_csr_matrix(method):
      # X can be a sparse matrix.
      rng = check_random_state(0)
      X = rng.randn(50, 2)
      X[(rng.randint(0, 50, 25), rng.randint(0, 2, 25))] = 0.0
      X_csr = sp.csr_matrix(X)
      tsne = TSNE(
          n_components=2,
          init="random",
          perplexity=10,
          learning_rate=100.0,
          random_state=0,
          method=method,
          n_iter=750,
      )
      X_embedded = tsne.fit_transform(X_csr)
      assert_allclose(trustworthiness(X_csr, X_embedded, n_neighbors=1), 1.0, rtol=1.1e-1)
  
  
  def test_preserve_trustworthiness_approximately_with_precomputed_distances():
      # Nearest neighbors should be preserved approximately.
      random_state = check_random_state(0)
      for i in range(3):
          X = random_state.randn(80, 2)
          D = squareform(pdist(X), "sqeuclidean")
          tsne = TSNE(
              n_components=2,
              perplexity=2,
              learning_rate=100.0,
              early_exaggeration=2.0,
              metric="precomputed",
              random_state=i,
              verbose=0,
              n_iter=500,
              init="random",
          )
          X_embedded = tsne.fit_transform(D)
          t = trustworthiness(D, X_embedded, n_neighbors=1, metric="precomputed")
          assert t > 0.95
  
  
  def test_trustworthiness_not_euclidean_metric():
      # Test trustworthiness with a metric different from 'euclidean' and
      # 'precomputed'
      random_state = check_random_state(0)
      X = random_state.randn(100, 2)
      assert trustworthiness(X, X, metric="cosine") == trustworthiness(
          pairwise_distances(X, metric="cosine"), X, metric="precomputed"
      )
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_early_exaggeration_too_small():
      # Early exaggeration factor must be >= 1.
      tsne = TSNE(early_exaggeration=0.99)
      with pytest.raises(ValueError, match="early_exaggeration .*"):
          tsne.fit_transform(np.array([[0.0], [0.0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_too_few_iterations():
      # Number of gradient descent iterations must be at least 200.
      tsne = TSNE(n_iter=199)
      with pytest.raises(ValueError, match="n_iter .*"):
          tsne.fit_transform(np.array([[0.0], [0.0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.parametrize(
      "method, retype",
      [
          ("exact", np.asarray),
          ("barnes_hut", np.asarray),
          ("barnes_hut", sp.csr_matrix),
      ],
  )
  @pytest.mark.parametrize(
      "D, message_regex",
      [
          ([[0.0], [1.0]], ".* square distance matrix"),
          ([[0.0, -1.0], [1.0, 0.0]], ".* positive.*"),
      ],
  )
  def test_bad_precomputed_distances(method, D, retype, message_regex):
      tsne = TSNE(
          metric="precomputed",
          method=method,
          init="random",
          random_state=42,
      )
      with pytest.raises(ValueError, match=message_regex):
          tsne.fit_transform(retype(D))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  def test_exact_no_precomputed_sparse():
      tsne = TSNE(
          metric="precomputed",
          method="exact",
          init="random",
          random_state=42,
      )
      with pytest.raises(TypeError, match="sparse"):
          tsne.fit_transform(sp.csr_matrix([[0, 5], [5, 0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  def test_high_perplexity_precomputed_sparse_distances():
      # Perplexity should be less than 50
      dist = np.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 0.0, 0.0]])
      bad_dist = sp.csr_matrix(dist)
      tsne = TSNE(metric="precomputed", init="random", random_state=42)
      msg = "3 neighbors per samples are required, but some samples have only 1"
      with pytest.raises(ValueError, match=msg):
          tsne.fit_transform(bad_dist)
  
  
  @ignore_warnings(category=EfficiencyWarning)
  def test_sparse_precomputed_distance():
      """Make sure that TSNE works identically for sparse and dense matrix"""
      random_state = check_random_state(0)
      X = random_state.randn(100, 2)
  
      D_sparse = kneighbors_graph(X, n_neighbors=100, mode="distance", include_self=True)
      D = pairwise_distances(X)
      assert sp.issparse(D_sparse)
      assert_almost_equal(D_sparse.A, D)
  
      tsne = TSNE(
          metric="precomputed", random_state=0, init="random", learning_rate="auto"
      )
      Xt_dense = tsne.fit_transform(D)
  
      for fmt in ["csr", "lil"]:
          Xt_sparse = tsne.fit_transform(D_sparse.asformat(fmt))
          assert_almost_equal(Xt_dense, Xt_sparse)
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_non_positive_computed_distances():
      # Computed distance matrices must be positive.
      def metric(x, y):
          return -1
  
      # Negative computed distances should be caught even if result is squared
      tsne = TSNE(metric=metric, method="exact")
      X = np.array([[0.0, 0.0], [1.0, 1.0]])
      with pytest.raises(ValueError, match="All distances .*metric given.*"):
          tsne.fit_transform(X)
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  def test_init_not_available():
      # 'init' must be 'pca', 'random', or numpy array.
      tsne = TSNE(init="not available")
      m = "'init' must be 'pca', 'random', or a numpy array"
      with pytest.raises(ValueError, match=m):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  def test_init_ndarray():
      # Initialize TSNE with ndarray and test fit
      tsne = TSNE(init=np.zeros((100, 2)), learning_rate="auto")
      X_embedded = tsne.fit_transform(np.ones((100, 5)))
      assert_array_equal(np.zeros((100, 2)), X_embedded)
  
  
  def test_init_ndarray_precomputed():
      # Initialize TSNE with ndarray and metric 'precomputed'
      # Make sure no FutureWarning is thrown from _fit
      tsne = TSNE(
          init=np.zeros((100, 2)),
          metric="precomputed",
          learning_rate=50.0,
      )
      tsne.fit(np.zeros((100, 100)))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_distance_not_available():
      # 'metric' must be valid.
      tsne = TSNE(metric="not available", method="exact")
      with pytest.raises(ValueError, match="Unknown metric not available.*"):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
      tsne = TSNE(metric="not available", method="barnes_hut")
      with pytest.raises(ValueError, match="Metric 'not available' not valid.*"):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_method_not_available():
      # 'nethod' must be 'barnes_hut' or 'exact'
      tsne = TSNE(method="not available")
      with pytest.raises(ValueError, match="'method' must be 'barnes_hut' or "):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_angle_out_of_range_checks():
      # check the angle parameter range
      for angle in [-1, -1e-6, 1 + 1e-6, 2]:
          tsne = TSNE(angle=angle)
          with pytest.raises(ValueError, match="'angle' must be between 0.0 - 1.0"):
              tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  def test_pca_initialization_not_compatible_with_precomputed_kernel():
      # Precomputed distance matrices cannot use PCA initialization.
      tsne = TSNE(metric="precomputed", init="pca")
      with pytest.raises(
          ValueError,
          match='The parameter init="pca" cannot be used with metric="precomputed".',
      ):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  def test_pca_initialization_not_compatible_with_sparse_input():
      # Sparse input matrices cannot use PCA initialization.
      tsne = TSNE(init="pca", learning_rate=100.0)
      with pytest.raises(TypeError, match="PCA initialization.*"):
          tsne.fit_transform(sp.csr_matrix([[0, 5], [5, 0]]))
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_n_components_range():
      # barnes_hut method should only be used with n_components <= 3
      tsne = TSNE(n_components=4, method="barnes_hut")
      with pytest.raises(ValueError, match="'n_components' should be .*"):
          tsne.fit_transform(np.array([[0.0], [1.0]]))
  
  
  def test_early_exaggeration_used():
      # check that the ``early_exaggeration`` parameter has an effect
      random_state = check_random_state(0)
      n_components = 2
      methods = ["exact", "barnes_hut"]
      X = random_state.randn(25, n_components).astype(np.float32)
      for method in methods:
          tsne = TSNE(
              n_components=n_components,
              perplexity=1,
              learning_rate=100.0,
              init="pca",
              random_state=0,
              method=method,
              early_exaggeration=1.0,
              n_iter=250,
          )
          X_embedded1 = tsne.fit_transform(X)
          tsne = TSNE(
              n_components=n_components,
              perplexity=1,
              learning_rate=100.0,
              init="pca",
              random_state=0,
              method=method,
              early_exaggeration=10.0,
              n_iter=250,
          )
          X_embedded2 = tsne.fit_transform(X)
  
          assert not np.allclose(X_embedded1, X_embedded2)
  
  
  def test_n_iter_used():
      # check that the ``n_iter`` parameter has an effect
      random_state = check_random_state(0)
      n_components = 2
      methods = ["exact", "barnes_hut"]
      X = random_state.randn(25, n_components).astype(np.float32)
      for method in methods:
          for n_iter in [251, 500]:
              tsne = TSNE(
                  n_components=n_components,
                  perplexity=1,
                  learning_rate=0.5,
                  init="random",
                  random_state=0,
                  method=method,
                  early_exaggeration=1.0,
                  n_iter=n_iter,
              )
              tsne.fit_transform(X)
  
              assert tsne.n_iter_ == n_iter - 1
  
  
  def test_answer_gradient_two_points():
      # Test the tree with only a single set of children.
      #
      # These tests & answers have been checked against the reference
      # implementation by LvdM.
      pos_input = np.array([[1.0, 0.0], [0.0, 1.0]])
      pos_output = np.array(
          [[-4.961291e-05, -1.072243e-04], [9.259460e-05, 2.702024e-04]]
      )
      neighbors = np.array([[1], [0]])
      grad_output = np.array(
          [[-2.37012478e-05, -6.29044398e-05], [2.37012478e-05, 6.29044398e-05]]
      )
      _run_answer_test(pos_input, pos_output, neighbors, grad_output)
  
  
  def test_answer_gradient_four_points():
      # Four points tests the tree with multiple levels of children.
      #
      # These tests & answers have been checked against the reference
      # implementation by LvdM.
      pos_input = np.array([[1.0, 0.0], [0.0, 1.0], [5.0, 2.0], [7.3, 2.2]])
      pos_output = np.array(
          [
              [6.080564e-05, -7.120823e-05],
              [-1.718945e-04, -4.000536e-05],
              [-2.271720e-04, 8.663310e-05],
              [-1.032577e-04, -3.582033e-05],
          ]
      )
      neighbors = np.array([[1, 2, 3], [0, 2, 3], [1, 0, 3], [1, 2, 0]])
      grad_output = np.array(
          [
              [5.81128448e-05, -7.78033454e-06],
              [-5.81526851e-05, 7.80976444e-06],
              [4.24275173e-08, -3.69569698e-08],
              [-2.58720939e-09, 7.52706374e-09],
          ]
      )
      _run_answer_test(pos_input, pos_output, neighbors, grad_output)
  
  
  def test_skip_num_points_gradient():
      # Test the kwargs option skip_num_points.
      #
      # Skip num points should make it such that the Barnes_hut gradient
      # is not calculated for indices below skip_num_point.
      # Aside from skip_num_points=2 and the first two gradient rows
      # being set to zero, these data points are the same as in
      # test_answer_gradient_four_points()
      pos_input = np.array([[1.0, 0.0], [0.0, 1.0], [5.0, 2.0], [7.3, 2.2]])
      pos_output = np.array(
          [
              [6.080564e-05, -7.120823e-05],
              [-1.718945e-04, -4.000536e-05],
              [-2.271720e-04, 8.663310e-05],
              [-1.032577e-04, -3.582033e-05],
          ]
      )
      neighbors = np.array([[1, 2, 3], [0, 2, 3], [1, 0, 3], [1, 2, 0]])
      grad_output = np.array(
          [
              [0.0, 0.0],
              [0.0, 0.0],
              [4.24275173e-08, -3.69569698e-08],
              [-2.58720939e-09, 7.52706374e-09],
          ]
      )
      _run_answer_test(pos_input, pos_output, neighbors, grad_output, False, 0.1, 2)
  
  
  def _run_answer_test(
      pos_input,
      pos_output,
      neighbors,
      grad_output,
      verbose=False,
      perplexity=0.1,
      skip_num_points=0,
  ):
      distances = pairwise_distances(pos_input).astype(np.float32)
      args = distances, perplexity, verbose
      pos_output = pos_output.astype(np.float32)
      neighbors = neighbors.astype(np.int64, copy=False)
      pij_input = _joint_probabilities(*args)
      pij_input = squareform(pij_input).astype(np.float32)
      grad_bh = np.zeros(pos_output.shape, dtype=np.float32)
  
      from scipy.sparse import csr_matrix
  
      P = csr_matrix(pij_input)
  
      neighbors = P.indices.astype(np.int64)
      indptr = P.indptr.astype(np.int64)
  
      _barnes_hut_tsne.gradient(
          P.data, pos_output, neighbors, indptr, grad_bh, 0.5, 2, 1, skip_num_points=0
      )
      assert_array_almost_equal(grad_bh, grad_output, decimal=4)
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_verbose():
      # Verbose options write to stdout.
      random_state = check_random_state(0)
      tsne = TSNE(verbose=2)
      X = random_state.randn(5, 2)
  
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          tsne.fit_transform(X)
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
  
      assert "[t-SNE]" in out
      assert "nearest neighbors..." in out
      assert "Computed conditional probabilities" in out
      assert "Mean sigma" in out
      assert "early exaggeration" in out
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_chebyshev_metric():
      # t-SNE should allow metrics that cannot be squared (issue #3526).
      random_state = check_random_state(0)
      tsne = TSNE(metric="chebyshev")
      X = random_state.randn(5, 2)
      tsne.fit_transform(X)
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_reduction_to_one_component():
      # t-SNE should allow reduction to one component (issue #4154).
      random_state = check_random_state(0)
      tsne = TSNE(n_components=1)
      X = random_state.randn(5, 2)
      X_embedded = tsne.fit(X).embedding_
      assert np.all(np.isfinite(X_embedded))
  
  
  @pytest.mark.parametrize("method", ["barnes_hut", "exact"])
  @pytest.mark.parametrize("dt", [np.float32, np.float64])
  def test_64bit(method, dt):
      # Ensure 64bit arrays are handled correctly.
      random_state = check_random_state(0)
  
      X = random_state.randn(10, 2).astype(dt, copy=False)
      tsne = TSNE(
          n_components=2,
          perplexity=2,
          learning_rate=100.0,
          random_state=0,
          method=method,
          verbose=0,
          n_iter=300,
          init="random",
      )
      X_embedded = tsne.fit_transform(X)
      effective_type = X_embedded.dtype
  
      # tsne cython code is only single precision, so the output will
      # always be single precision, irrespectively of the input dtype
      assert effective_type == np.float32
  
  
  @pytest.mark.parametrize("method", ["barnes_hut", "exact"])
  def test_kl_divergence_not_nan(method):
      # Ensure kl_divergence_ is computed at last iteration
      # even though n_iter % n_iter_check != 0, i.e. 1003 % 50 != 0
      random_state = check_random_state(0)
  
      X = random_state.randn(50, 2)
      tsne = TSNE(
          n_components=2,
          perplexity=2,
          learning_rate=100.0,
          random_state=0,
          method=method,
          verbose=0,
          n_iter=503,
          init="random",
      )
      tsne.fit_transform(X)
  
      assert not np.isnan(tsne.kl_divergence_)
  
  
  def test_barnes_hut_angle():
      # When Barnes-Hut's angle=0 this corresponds to the exact method.
      angle = 0.0
      perplexity = 10
      n_samples = 100
      for n_components in [2, 3]:
          n_features = 5
          degrees_of_freedom = float(n_components - 1.0)
  
          random_state = check_random_state(0)
          data = random_state.randn(n_samples, n_features)
          distances = pairwise_distances(data)
          params = random_state.randn(n_samples, n_components)
          P = _joint_probabilities(distances, perplexity, verbose=0)
          kl_exact, grad_exact = _kl_divergence(
              params, P, degrees_of_freedom, n_samples, n_components
          )
  
          n_neighbors = n_samples - 1
          distances_csr = (
              NearestNeighbors()
              .fit(data)
              .kneighbors_graph(n_neighbors=n_neighbors, mode="distance")
          )
          P_bh = _joint_probabilities_nn(distances_csr, perplexity, verbose=0)
          kl_bh, grad_bh = _kl_divergence_bh(
              params,
              P_bh,
              degrees_of_freedom,
              n_samples,
              n_components,
              angle=angle,
              skip_num_points=0,
              verbose=0,
          )
  
          P = squareform(P)
          P_bh = P_bh.toarray()
          assert_array_almost_equal(P_bh, P, decimal=5)
          assert_almost_equal(kl_exact, kl_bh, decimal=3)
  
  
  @skip_if_32bit
  def test_n_iter_without_progress():
      # Use a dummy negative n_iter_without_progress and check output on stdout
      random_state = check_random_state(0)
      X = random_state.randn(100, 10)
      for method in ["barnes_hut", "exact"]:
          tsne = TSNE(
              n_iter_without_progress=-1,
              verbose=2,
              learning_rate=1e8,
              random_state=0,
              method=method,
              n_iter=351,
              init="random",
          )
          tsne._N_ITER_CHECK = 1
          tsne._EXPLORATION_N_ITER = 0
  
          old_stdout = sys.stdout
          sys.stdout = StringIO()
          try:
              tsne.fit_transform(X)
          finally:
              out = sys.stdout.getvalue()
              sys.stdout.close()
              sys.stdout = old_stdout
  
          # The output needs to contain the value of n_iter_without_progress
          assert "did not make any progress during the last -1 episodes. Finished." in out
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_min_grad_norm():
      # Make sure that the parameter min_grad_norm is used correctly
      random_state = check_random_state(0)
      X = random_state.randn(100, 2)
      min_grad_norm = 0.002
      tsne = TSNE(min_grad_norm=min_grad_norm, verbose=2, random_state=0, method="exact")
  
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          tsne.fit_transform(X)
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
  
      lines_out = out.split("\n")
  
      # extract the gradient norm from the verbose output
      gradient_norm_values = []
      for line in lines_out:
          # When the computation is Finished just an old gradient norm value
          # is repeated that we do not need to store
          if "Finished" in line:
              break
  
          start_grad_norm = line.find("gradient norm")
          if start_grad_norm >= 0:
              line = line[start_grad_norm:]
              line = line.replace("gradient norm = ", "").split(" ")[0]
              gradient_norm_values.append(float(line))
  
      # Compute how often the gradient norm is smaller than min_grad_norm
      gradient_norm_values = np.array(gradient_norm_values)
      n_smaller_gradient_norms = len(
          gradient_norm_values[gradient_norm_values <= min_grad_norm]
      )
  
      # The gradient norm can be smaller than min_grad_norm at most once,
      # because in the moment it becomes smaller the optimization stops
      assert n_smaller_gradient_norms <= 1
  
  
  @pytest.mark.filterwarnings("ignore:The default learning rate in TSNE")
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_accessible_kl_divergence():
      # Ensures that the accessible kl_divergence matches the computed value
      random_state = check_random_state(0)
      X = random_state.randn(50, 2)
      tsne = TSNE(
          n_iter_without_progress=2, verbose=2, random_state=0, method="exact", n_iter=500
      )
  
      old_stdout = sys.stdout
      sys.stdout = StringIO()
      try:
          tsne.fit_transform(X)
      finally:
          out = sys.stdout.getvalue()
          sys.stdout.close()
          sys.stdout = old_stdout
  
      # The output needs to contain the accessible kl_divergence as the error at
      # the last iteration
      for line in out.split("\n")[::-1]:
          if "Iteration" in line:
              _, _, error = line.partition("error = ")
              if error:
                  error, _, _ = error.partition(",")
                  break
      assert_almost_equal(tsne.kl_divergence_, float(error), decimal=5)
  
  
  @pytest.mark.parametrize("method", ["barnes_hut", "exact"])
  def test_uniform_grid(method):
      """Make sure that TSNE can approximately recover a uniform 2D grid
  
      Due to ties in distances between point in X_2d_grid, this test is platform
      dependent for ``method='barnes_hut'`` due to numerical imprecision.
  
      Also, t-SNE is not assured to converge to the right solution because bad
      initialization can lead to convergence to bad local minimum (the
      optimization problem is non-convex). To avoid breaking the test too often,
      we re-run t-SNE from the final point when the convergence is not good
      enough.
      """
      seeds = range(3)
      n_iter = 500
      for seed in seeds:
          tsne = TSNE(
              n_components=2,
              init="random",
              random_state=seed,
              perplexity=50,
              n_iter=n_iter,
              method=method,
              learning_rate="auto",
          )
          Y = tsne.fit_transform(X_2d_grid)
  
          try_name = "{}_{}".format(method, seed)
          try:
              assert_uniform_grid(Y, try_name)
          except AssertionError:
              # If the test fails a first time, re-run with init=Y to see if
              # this was caused by a bad initialization. Note that this will
              # also run an early_exaggeration step.
              try_name += ":rerun"
              tsne.init = Y
              Y = tsne.fit_transform(X_2d_grid)
              assert_uniform_grid(Y, try_name)
  
  
  def assert_uniform_grid(Y, try_name=None):
      # Ensure that the resulting embedding leads to approximately
      # uniformly spaced points: the distance to the closest neighbors
      # should be non-zero and approximately constant.
      nn = NearestNeighbors(n_neighbors=1).fit(Y)
      dist_to_nn = nn.kneighbors(return_distance=True)[0].ravel()
      assert dist_to_nn.min() > 0.1
  
      smallest_to_mean = dist_to_nn.min() / np.mean(dist_to_nn)
      largest_to_mean = dist_to_nn.max() / np.mean(dist_to_nn)
  
      assert smallest_to_mean > 0.5, try_name
      assert largest_to_mean < 2, try_name
  
  
  def test_bh_match_exact():
      # check that the ``barnes_hut`` method match the exact one when
      # ``angle = 0`` and ``perplexity > n_samples / 3``
      random_state = check_random_state(0)
      n_features = 10
      X = random_state.randn(30, n_features).astype(np.float32)
      X_embeddeds = {}
      n_iter = {}
      for method in ["exact", "barnes_hut"]:
          tsne = TSNE(
              n_components=2,
              method=method,
              learning_rate=1.0,
              init="random",
              random_state=0,
              n_iter=251,
              perplexity=30.0,
              angle=0,
          )
          # Kill the early_exaggeration
          tsne._EXPLORATION_N_ITER = 0
          X_embeddeds[method] = tsne.fit_transform(X)
          n_iter[method] = tsne.n_iter_
  
      assert n_iter["exact"] == n_iter["barnes_hut"]
      assert_allclose(X_embeddeds["exact"], X_embeddeds["barnes_hut"], rtol=1e-4)
  
  
  def test_gradient_bh_multithread_match_sequential():
      # check that the bh gradient with different num_threads gives the same
      # results
  
      n_features = 10
      n_samples = 30
      n_components = 2
      degrees_of_freedom = 1
  
      angle = 3
      perplexity = 5
  
      random_state = check_random_state(0)
      data = random_state.randn(n_samples, n_features).astype(np.float32)
      params = random_state.randn(n_samples, n_components)
  
      n_neighbors = n_samples - 1
      distances_csr = (
          NearestNeighbors()
          .fit(data)
          .kneighbors_graph(n_neighbors=n_neighbors, mode="distance")
      )
      P_bh = _joint_probabilities_nn(distances_csr, perplexity, verbose=0)
      kl_sequential, grad_sequential = _kl_divergence_bh(
          params,
          P_bh,
          degrees_of_freedom,
          n_samples,
          n_components,
          angle=angle,
          skip_num_points=0,
          verbose=0,
          num_threads=1,
      )
      for num_threads in [2, 4]:
          kl_multithread, grad_multithread = _kl_divergence_bh(
              params,
              P_bh,
              degrees_of_freedom,
              n_samples,
              n_components,
              angle=angle,
              skip_num_points=0,
              verbose=0,
              num_threads=num_threads,
          )
  
          assert_allclose(kl_multithread, kl_sequential, rtol=1e-6)
          assert_allclose(grad_multithread, grad_multithread)
  
  
  def test_tsne_with_different_distance_metrics():
      """Make sure that TSNE works for different distance metrics"""
      random_state = check_random_state(0)
      n_components_original = 3
      n_components_embedding = 2
      X = random_state.randn(50, n_components_original).astype(np.float32)
      metrics = ["manhattan", "cosine"]
      dist_funcs = [manhattan_distances, cosine_distances]
      for metric, dist_func in zip(metrics, dist_funcs):
          X_transformed_tsne = TSNE(
              metric=metric,
              n_components=n_components_embedding,
              random_state=0,
              n_iter=300,
              init="random",
              learning_rate="auto",
          ).fit_transform(X)
          X_transformed_tsne_precomputed = TSNE(
              metric="precomputed",
              n_components=n_components_embedding,
              random_state=0,
              n_iter=300,
              init="random",
              learning_rate="auto",
          ).fit_transform(dist_func(X))
          assert_array_equal(X_transformed_tsne, X_transformed_tsne_precomputed)
  
  
  # TODO: Remove in 1.2
  @pytest.mark.parametrize("init", [None, "random", "pca"])
  def test_tsne_init_futurewarning(init):
      """Make sure that a FutureWarning is only raised when the
      init is not specified or is 'pca'."""
      random_state = check_random_state(0)
  
      X = random_state.randn(5, 2)
      kwargs = dict(learning_rate=200.0, init=init)
      tsne = TSNE(**{k: v for k, v in kwargs.items() if v is not None})
  
      if init is None:
          with pytest.warns(FutureWarning, match="The default initialization.*"):
              tsne.fit_transform(X)
      elif init == "pca":
          with pytest.warns(FutureWarning, match="The PCA initialization.*"):
              tsne.fit_transform(X)
      else:
          with pytest.warns(None) as record:
              tsne.fit_transform(X)
          assert not record
  
  
  # TODO: Remove in 1.2
  @pytest.mark.parametrize("learning_rate", [None, 200.0])
  def test_tsne_learning_rate_futurewarning(learning_rate):
      """Make sure that a FutureWarning is only raised when the learning rate
      is not specified"""
      random_state = check_random_state(0)
  
      X = random_state.randn(5, 2)
      kwargs = dict(learning_rate=learning_rate, init="random")
      tsne = TSNE(**{k: v for k, v in kwargs.items() if v is not None})
  
      if learning_rate is None:
          with pytest.warns(FutureWarning, match="The default learning rate.*"):
              tsne.fit_transform(X)
      else:
          with pytest.warns(None) as record:
              tsne.fit_transform(X)
          assert not record
  
  
  @pytest.mark.filterwarnings("ignore:The default initialization in TSNE")
  def test_tsne_negative_learning_rate():
      """Make sure that negative learning rate results in a ValueError"""
      random_state = check_random_state(0)
      X = random_state.randn(5, 2)
      with pytest.raises(ValueError, match="'learning_rate' must be.*"):
          TSNE(learning_rate=-50.0).fit_transform(X)
  
  
  @pytest.mark.parametrize("method", ["exact", "barnes_hut"])
  def test_tsne_n_jobs(method):
      """Make sure that the n_jobs parameter doesn't impact the output"""
      random_state = check_random_state(0)
      n_features = 10
      X = random_state.randn(30, n_features)
      X_tr_ref = TSNE(
          n_components=2,
          method=method,
          perplexity=30.0,
          angle=0,
          n_jobs=1,
          random_state=0,
          init="random",
          learning_rate="auto",
      ).fit_transform(X)
      X_tr = TSNE(
          n_components=2,
          method=method,
          perplexity=30.0,
          angle=0,
          n_jobs=2,
          random_state=0,
          init="random",
          learning_rate="auto",
      ).fit_transform(X)
  
      assert_allclose(X_tr_ref, X_tr)
  
  
  @pytest.mark.filterwarnings("ignore:The PCA initialization in TSNE will change")
  # FIXME: remove in 1.3 after deprecation of `square_distances`
  def test_tsne_deprecation_square_distances():
      """Check that we raise a warning regarding the removal of
      `square_distances`.
  
      Also check the parameters do not have any effect.
      """
      random_state = check_random_state(0)
      X = random_state.randn(30, 10)
      tsne = TSNE(
          n_components=2,
          init="pca",
          learning_rate="auto",
          perplexity=30.0,
          angle=0,
          n_jobs=1,
          random_state=0,
          square_distances=True,
      )
      warn_msg = (
          "The parameter `square_distances` has not effect and will be removed in"
          " version 1.3"
      )
      with pytest.warns(FutureWarning, match=warn_msg):
          X_trans_1 = tsne.fit_transform(X)
  
      tsne = TSNE(
          n_components=2,
          init="pca",
          learning_rate="auto",
          perplexity=30.0,
          angle=0,
          n_jobs=1,
          random_state=0,
      )
      X_trans_2 = tsne.fit_transform(X)
      assert_allclose(X_trans_1, X_trans_2)
  
apis: numpy.ones scipy.spatial.distance.pdist sklearn.metrics.pairwise.pairwise_distances sklearn.manifold._t_sne._kl_divergence_bh numpy.log numpy.fill_diagonal sklearn.manifold._barnes_hut_tsne.gradient numpy.meshgrid numpy.isfinite sklearn.manifold._t_sne._joint_probabilities_nn numpy.allclose sklearn.neighbors.NearestNeighbors sklearn.utils._testing.assert_array_equal sklearn.manifold.TSNE sklearn.manifold._t_sne.trustworthiness numpy.isnan numpy.linspace numpy.mean sklearn.utils._testing.assert_array_almost_equal sklearn.utils.check_random_state numpy.zeros numpy.arange sklearn.manifold._utils._binary_search_perplexity sklearn.utils._testing.ignore_warnings numpy.finfo sklearn.datasets.make_blobs scipy.spatial.distance.squareform numpy.log2 sklearn.manifold._t_sne._joint_probabilities scipy.sparse.issparse sklearn.utils._testing.assert_almost_equal scipy.sparse.csr_matrix sklearn.neighbors.kneighbors_graph sklearn.manifold._t_sne._kl_divergence numpy.testing.assert_allclose numpy.array