files: import numpy as np
  import numba
  import umap.distances as dist
  from umap.utils import tau_rand_int
  
  
  @numba.njit()
  def clip(val):
      """Standard clamping of a value into a fixed range (in this case -4.0 to
      4.0)
  
      Parameters
      ----------
      val: float
          The value to be clamped.
  
      Returns
      -------
      The clamped value, now fixed to be in the range -4.0 to 4.0.
      """
      if val > 4.0:
          return 4.0
      elif val < -4.0:
          return -4.0
      else:
          return val
  
  
  @numba.njit(
      "f4(f4[::1],f4[::1])",
      fastmath=True,
      cache=True,
      locals={
          "result": numba.types.float32,
          "diff": numba.types.float32,
          "dim": numba.types.intp,
      },
  )
  def rdist(x, y):
      """Reduced Euclidean distance.
  
      Parameters
      ----------
      x: array of shape (embedding_dim,)
      y: array of shape (embedding_dim,)
  
      Returns
      -------
      The squared euclidean distance between x and y
      """
      result = 0.0
      dim = x.shape[0]
      for i in range(dim):
          diff = x[i] - y[i]
          result += diff * diff
  
      return result
  
  
  def _optimize_layout_euclidean_single_epoch(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma,
      dim,
      move_other,
      alpha,
      epochs_per_negative_sample,
      epoch_of_next_negative_sample,
      epoch_of_next_sample,
      n,
      densmap_flag,
      dens_phi_sum,
      dens_re_sum,
      dens_re_cov,
      dens_re_std,
      dens_re_mean,
      dens_lambda,
      dens_R,
      dens_mu,
      dens_mu_tot,
  ):
      for i in numba.prange(epochs_per_sample.shape[0]):
          if epoch_of_next_sample[i] <= n:
              j = head[i]
              k = tail[i]
  
              current = head_embedding[j]
              other = tail_embedding[k]
  
              dist_squared = rdist(current, other)
  
              if densmap_flag:
                  phi = 1.0 / (1.0 + a * pow(dist_squared, b))
                  dphi_term = (
                      a * b * pow(dist_squared, b - 1) / (1.0 + a * pow(dist_squared, b))
                  )
  
                  q_jk = phi / dens_phi_sum[k]
                  q_kj = phi / dens_phi_sum[j]
  
                  drk = q_jk * (
                      (1.0 - b * (1 - phi)) / np.exp(dens_re_sum[k]) + dphi_term
                  )
                  drj = q_kj * (
                      (1.0 - b * (1 - phi)) / np.exp(dens_re_sum[j]) + dphi_term
                  )
  
                  re_std_sq = dens_re_std * dens_re_std
                  weight_k = (
                      dens_R[k]
                      - dens_re_cov * (dens_re_sum[k] - dens_re_mean) / re_std_sq
                  )
                  weight_j = (
                      dens_R[j]
                      - dens_re_cov * (dens_re_sum[j] - dens_re_mean) / re_std_sq
                  )
  
                  grad_cor_coeff = (
                      dens_lambda
                      * dens_mu_tot
                      * (weight_k * drk + weight_j * drj)
                      / (dens_mu[i] * dens_re_std)
                      / n_vertices
                  )
  
              if dist_squared > 0.0:
                  grad_coeff = -2.0 * a * b * pow(dist_squared, b - 1.0)
                  grad_coeff /= a * pow(dist_squared, b) + 1.0
              else:
                  grad_coeff = 0.0
  
              for d in range(dim):
                  grad_d = clip(grad_coeff * (current[d] - other[d]))
  
                  if densmap_flag:
                      grad_d += clip(2 * grad_cor_coeff * (current[d] - other[d]))
  
                  current[d] += grad_d * alpha
                  if move_other:
                      other[d] += -grad_d * alpha
  
              epoch_of_next_sample[i] += epochs_per_sample[i]
  
              n_neg_samples = int(
                  (n - epoch_of_next_negative_sample[i]) / epochs_per_negative_sample[i]
              )
  
              for p in range(n_neg_samples):
                  k = tau_rand_int(rng_state) % n_vertices
  
                  other = tail_embedding[k]
  
                  dist_squared = rdist(current, other)
  
                  if dist_squared > 0.0:
                      grad_coeff = 2.0 * gamma * b
                      grad_coeff /= (0.001 + dist_squared) * (
                          a * pow(dist_squared, b) + 1
                      )
                  elif j == k:
                      continue
                  else:
                      grad_coeff = 0.0
  
                  for d in range(dim):
                      if grad_coeff > 0.0:
                          grad_d = clip(grad_coeff * (current[d] - other[d]))
                      else:
                          grad_d = 4.0
                      current[d] += grad_d * alpha
  
              epoch_of_next_negative_sample[i] += (
                  n_neg_samples * epochs_per_negative_sample[i]
              )
  
  
  def _optimize_layout_euclidean_densmap_epoch_init(
      head_embedding,
      tail_embedding,
      head,
      tail,
      a,
      b,
      re_sum,
      phi_sum,
  ):
      re_sum.fill(0)
      phi_sum.fill(0)
  
      for i in numba.prange(head.size):
          j = head[i]
          k = tail[i]
  
          current = head_embedding[j]
          other = tail_embedding[k]
          dist_squared = rdist(current, other)
  
          phi = 1.0 / (1.0 + a * pow(dist_squared, b))
  
          re_sum[j] += phi * dist_squared
          re_sum[k] += phi * dist_squared
          phi_sum[j] += phi
          phi_sum[k] += phi
  
      epsilon = 1e-8
      for i in range(re_sum.size):
          re_sum[i] = np.log(epsilon + (re_sum[i] / phi_sum[i]))
  
  
  def optimize_layout_euclidean(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      parallel=False,
      verbose=False,
      densmap=False,
      densmap_kwds={},
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
      n_epochs: int, or list of int
          The number of training epochs to use in optimization, or a list of
          epochs at which to save the embedding. In case of a list, the optimization
          will use the maximum number of epochs in the list, and will return a list
          of embedding in the order of increasing epoch, regardless of the order in
          the epoch list.
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
      epochs_per_samples: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
      a: float
          Parameter of differentiable approximation of right adjoint functor
      b: float
          Parameter of differentiable approximation of right adjoint functor
      rng_state: array of int64, shape (3,)
          The internal state of the rng
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
      parallel: bool (optional, default False)
          Whether to run the computation using numba parallel.
          Running in parallel is non-deterministic, and is not used
          if a random seed has been set, to ensure reproducibility.
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
      densmap: bool (optional, default False)
          Whether to use the density-augmented densMAP objective
      densmap_kwds: dict (optional, default {})
          Auxiliary data for densMAP
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      optimize_fn = numba.njit(
          _optimize_layout_euclidean_single_epoch, fastmath=True, parallel=parallel
      )
  
      if densmap:
          dens_init_fn = numba.njit(
              _optimize_layout_euclidean_densmap_epoch_init,
              fastmath=True,
              parallel=parallel,
          )
  
          dens_mu_tot = np.sum(densmap_kwds["mu_sum"]) / 2
          dens_lambda = densmap_kwds["lambda"]
          dens_R = densmap_kwds["R"]
          dens_mu = densmap_kwds["mu"]
          dens_phi_sum = np.zeros(n_vertices, dtype=np.float32)
          dens_re_sum = np.zeros(n_vertices, dtype=np.float32)
          dens_var_shift = densmap_kwds["var_shift"]
      else:
          dens_mu_tot = 0
          dens_lambda = 0
          dens_R = np.zeros(1, dtype=np.float32)
          dens_mu = np.zeros(1, dtype=np.float32)
          dens_phi_sum = np.zeros(1, dtype=np.float32)
          dens_re_sum = np.zeros(1, dtype=np.float32)
  
      epochs_list = None
      embedding_list = []
      if isinstance(n_epochs, list):
          epochs_list = n_epochs
          n_epochs = max(epochs_list)
  
      for n in range(n_epochs):
  
          densmap_flag = (
              densmap
              and (densmap_kwds["lambda"] > 0)
              and (((n + 1) / float(n_epochs)) > (1 - densmap_kwds["frac"]))
          )
  
          if densmap_flag:
              dens_init_fn(
                  head_embedding,
                  tail_embedding,
                  head,
                  tail,
                  a,
                  b,
                  dens_re_sum,
                  dens_phi_sum,
              )
  
              dens_re_std = np.sqrt(np.var(dens_re_sum) + dens_var_shift)
              dens_re_mean = np.mean(dens_re_sum)
              dens_re_cov = np.dot(dens_re_sum, dens_R) / (n_vertices - 1)
          else:
              dens_re_std = 0
              dens_re_mean = 0
              dens_re_cov = 0
  
          optimize_fn(
              head_embedding,
              tail_embedding,
              head,
              tail,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              dim,
              move_other,
              alpha,
              epochs_per_negative_sample,
              epoch_of_next_negative_sample,
              epoch_of_next_sample,
              n,
              densmap_flag,
              dens_phi_sum,
              dens_re_sum,
              dens_re_cov,
              dens_re_std,
              dens_re_mean,
              dens_lambda,
              dens_R,
              dens_mu,
              dens_mu_tot,
          )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
          if epochs_list is not None and n in epochs_list:
              embedding_list.append(head_embedding.copy())
  
      # Add the last embedding to the list as well
      if epochs_list is not None:
          embedding_list.append(head_embedding.copy())
  
      return head_embedding if epochs_list is None else embedding_list
  
  
  @numba.njit(fastmath=True)
  def optimize_layout_generic(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      output_metric=dist.euclidean,
      output_metric_kwds=(),
      verbose=False,
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
  
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
  
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
  
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
  
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
  
      weight: array of shape (n_1_simplices)
          The membership weights of the 1-simplices.
  
      n_epochs: int
          The number of training epochs to use in optimization.
  
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
  
      epochs_per_sample: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      rng_state: array of int64, shape (3,)
          The internal state of the rng
  
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
  
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
  
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      for n in range(n_epochs):
          for i in range(epochs_per_sample.shape[0]):
              if epoch_of_next_sample[i] <= n:
                  j = head[i]
                  k = tail[i]
  
                  current = head_embedding[j]
                  other = tail_embedding[k]
  
                  dist_output, grad_dist_output = output_metric(
                      current, other, *output_metric_kwds
                  )
                  _, rev_grad_dist_output = output_metric(
                      other, current, *output_metric_kwds
                  )
  
                  if dist_output > 0.0:
                      w_l = pow((1 + a * pow(dist_output, 2 * b)), -1)
                  else:
                      w_l = 1.0
                  grad_coeff = 2 * b * (w_l - 1) / (dist_output + 1e-6)
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * grad_dist_output[d])
  
                      current[d] += grad_d * alpha
                      if move_other:
                          grad_d = clip(grad_coeff * rev_grad_dist_output[d])
                          other[d] += grad_d * alpha
  
                  epoch_of_next_sample[i] += epochs_per_sample[i]
  
                  n_neg_samples = int(
                      (n - epoch_of_next_negative_sample[i])
                      / epochs_per_negative_sample[i]
                  )
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % n_vertices
  
                      other = tail_embedding[k]
  
                      dist_output, grad_dist_output = output_metric(
                          current, other, *output_metric_kwds
                      )
  
                      if dist_output > 0.0:
                          w_l = pow((1 + a * pow(dist_output, 2 * b)), -1)
                      elif j == k:
                          continue
                      else:
                          w_l = 1.0
  
                      grad_coeff = gamma * 2 * b * w_l / (dist_output + 1e-6)
  
                      for d in range(dim):
                          grad_d = clip(grad_coeff * grad_dist_output[d])
                          current[d] += grad_d * alpha
  
                  epoch_of_next_negative_sample[i] += (
                      n_neg_samples * epochs_per_negative_sample[i]
                  )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embedding
  
  
  @numba.njit(fastmath=True)
  def optimize_layout_inverse(
      head_embedding,
      tail_embedding,
      head,
      tail,
      weight,
      sigmas,
      rhos,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      output_metric=dist.euclidean,
      output_metric_kwds=(),
      verbose=False,
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
  
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
  
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
  
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
  
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
  
      weight: array of shape (n_1_simplices)
          The membership weights of the 1-simplices.
  
      n_epochs: int
          The number of training epochs to use in optimization.
  
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
  
      epochs_per_sample: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      rng_state: array of int64, shape (3,)
          The internal state of the rng
  
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
  
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
  
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      for n in range(n_epochs):
          for i in range(epochs_per_sample.shape[0]):
              if epoch_of_next_sample[i] <= n:
                  j = head[i]
                  k = tail[i]
  
                  current = head_embedding[j]
                  other = tail_embedding[k]
  
                  dist_output, grad_dist_output = output_metric(
                      current, other, *output_metric_kwds
                  )
  
                  w_l = weight[i]
                  grad_coeff = -(1 / (w_l * sigmas[k] + 1e-6))
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * grad_dist_output[d])
  
                      current[d] += grad_d * alpha
                      if move_other:
                          other[d] += -grad_d * alpha
  
                  epoch_of_next_sample[i] += epochs_per_sample[i]
  
                  n_neg_samples = int(
                      (n - epoch_of_next_negative_sample[i])
                      / epochs_per_negative_sample[i]
                  )
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % n_vertices
  
                      other = tail_embedding[k]
  
                      dist_output, grad_dist_output = output_metric(
                          current, other, *output_metric_kwds
                      )
  
                      # w_l = 0.0 # for negative samples, the edge does not exist
                      w_h = np.exp(-max(dist_output - rhos[k], 1e-6) / (sigmas[k] + 1e-6))
                      grad_coeff = -gamma * ((0 - w_h) / ((1 - w_h) * sigmas[k] + 1e-6))
  
                      for d in range(dim):
                          grad_d = clip(grad_coeff * grad_dist_output[d])
                          current[d] += grad_d * alpha
  
                  epoch_of_next_negative_sample[i] += (
                      n_neg_samples * epochs_per_negative_sample[i]
                  )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embedding
  
  
  def _optimize_layout_aligned_euclidean_single_epoch(
      head_embeddings,
      tail_embeddings,
      heads,
      tails,
      epochs_per_sample,
      a,
      b,
      regularisation_weights,
      relations,
      rng_state,
      gamma,
      lambda_,
      dim,
      move_other,
      alpha,
      epochs_per_negative_sample,
      epoch_of_next_negative_sample,
      epoch_of_next_sample,
      n,
  ):
      n_embeddings = len(heads)
      window_size = (relations.shape[1] - 1) // 2
  
      max_n_edges = 0
      for e_p_s in epochs_per_sample:
          if e_p_s.shape[0] >= max_n_edges:
              max_n_edges = e_p_s.shape[0]
  
      embedding_order = np.arange(n_embeddings).astype(np.int32)
      np.random.seed(abs(rng_state[0]))
      np.random.shuffle(embedding_order)
  
      for i in range(max_n_edges):
          for m in embedding_order:
              if i < epoch_of_next_sample[m].shape[0] and epoch_of_next_sample[m][i] <= n:
                  j = heads[m][i]
                  k = tails[m][i]
  
                  current = head_embeddings[m][j]
                  other = tail_embeddings[m][k]
  
                  dist_squared = rdist(current, other)
  
                  if dist_squared > 0.0:
                      grad_coeff = -2.0 * a * b * pow(dist_squared, b - 1.0)
                      grad_coeff /= a * pow(dist_squared, b) + 1.0
                  else:
                      grad_coeff = 0.0
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * (current[d] - other[d]))
  
                      for offset in range(-window_size, window_size):
                          neighbor_m = m + offset
                          if (
                              neighbor_m >= 0
                              and neighbor_m < n_embeddings
                              and offset != 0
                          ):
                              identified_index = relations[m, offset + window_size, j]
                              if identified_index >= 0:
                                  grad_d -= clip(
                                      (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                      * regularisation_weights[m, offset + window_size, j]
                                      * (
                                          current[d]
                                          - head_embeddings[neighbor_m][
                                              identified_index, d
                                          ]
                                      )
                                  )
  
                      current[d] += clip(grad_d) * alpha
                      if move_other:
                          other_grad_d = clip(grad_coeff * (other[d] - current[d]))
  
                          for offset in range(-window_size, window_size):
                              neighbor_m = m + offset
                              if (
                                  neighbor_m >= 0
                                  and neighbor_m < n_embeddings
                                  and offset != 0
                              ):
                                  identified_index = relations[m, offset + window_size, k]
                                  if identified_index >= 0:
                                      grad_d -= clip(
                                          (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                          * regularisation_weights[
                                              m, offset + window_size, k
                                          ]
                                          * (
                                              other[d]
                                              - head_embeddings[neighbor_m][
                                                  identified_index, d
                                              ]
                                          )
                                      )
  
                          other[d] += clip(other_grad_d) * alpha
  
                  epoch_of_next_sample[m][i] += epochs_per_sample[m][i]
  
                  if epochs_per_negative_sample[m][i] > 0:
                      n_neg_samples = int(
                          (n - epoch_of_next_negative_sample[m][i])
                          / epochs_per_negative_sample[m][i]
                      )
                  else:
                      n_neg_samples = 0
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % tail_embeddings[m].shape[0]
  
                      other = tail_embeddings[m][k]
  
                      dist_squared = rdist(current, other)
  
                      if dist_squared > 0.0:
                          grad_coeff = 2.0 * gamma * b
                          grad_coeff /= (0.001 + dist_squared) * (
                              a * pow(dist_squared, b) + 1
                          )
                      elif j == k:
                          continue
                      else:
                          grad_coeff = 0.0
  
                      for d in range(dim):
                          if grad_coeff > 0.0:
                              grad_d = clip(grad_coeff * (current[d] - other[d]))
                          else:
                              grad_d = 4.0
  
                          for offset in range(-window_size, window_size):
                              neighbor_m = m + offset
                              if (
                                  neighbor_m >= 0
                                  and neighbor_m < n_embeddings
                                  and offset != 0
                              ):
                                  identified_index = relations[m, offset + window_size, j]
                                  if identified_index >= 0:
                                      grad_d -= clip(
                                          (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                          * regularisation_weights[
                                              m, offset + window_size, j
                                          ]
                                          * (
                                              current[d]
                                              - head_embeddings[neighbor_m][
                                                  identified_index, d
                                              ]
                                          )
                                      )
  
                          current[d] += clip(grad_d) * alpha
  
                  epoch_of_next_negative_sample[m][i] += (
                      n_neg_samples * epochs_per_negative_sample[m][i]
                  )
  
  
  def optimize_layout_aligned_euclidean(
      head_embeddings,
      tail_embeddings,
      heads,
      tails,
      n_epochs,
      epochs_per_sample,
      regularisation_weights,
      relations,
      rng_state,
      a=1.576943460405378,
      b=0.8950608781227859,
      gamma=1.0,
      lambda_=5e-3,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      parallel=True,
      verbose=False,
      move_other=False,
  ):
      dim = head_embeddings[0].shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = numba.typed.List.empty_list(numba.types.float32[::1])
      epoch_of_next_negative_sample = numba.typed.List.empty_list(
          numba.types.float32[::1]
      )
      epoch_of_next_sample = numba.typed.List.empty_list(numba.types.float32[::1])
  
      for m in range(len(heads)):
          epochs_per_negative_sample.append(
              epochs_per_sample[m].astype(np.float32) / negative_sample_rate
          )
          epoch_of_next_negative_sample.append(
              epochs_per_negative_sample[m].astype(np.float32)
          )
          epoch_of_next_sample.append(epochs_per_sample[m].astype(np.float32))
  
      optimize_fn = numba.njit(
          _optimize_layout_aligned_euclidean_single_epoch,
          fastmath=True,
          parallel=parallel,
      )
  
      for n in range(n_epochs):
          optimize_fn(
              head_embeddings,
              tail_embeddings,
              heads,
              tails,
              epochs_per_sample,
              a,
              b,
              regularisation_weights,
              relations,
              rng_state,
              gamma,
              lambda_,
              dim,
              move_other,
              alpha,
              epochs_per_negative_sample,
              epoch_of_next_negative_sample,
              epoch_of_next_sample,
              n,
          )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embeddings
  
files: umap/layouts.py
files: import numpy as np
  import numba
  import umap.distances as dist
  from umap.utils import tau_rand_int
  
  
  @numba.njit()
  def clip(val):
      """Standard clamping of a value into a fixed range (in this case -4.0 to
      4.0)
  
      Parameters
      ----------
      val: float
          The value to be clamped.
  
      Returns
      -------
      The clamped value, now fixed to be in the range -4.0 to 4.0.
      """
      if val > 4.0:
          return 4.0
      elif val < -4.0:
          return -4.0
      else:
          return val
  
  
  @numba.njit(
      "f4(f4[::1],f4[::1])",
      fastmath=True,
      cache=True,
      locals={
          "result": numba.types.float32,
          "diff": numba.types.float32,
          "dim": numba.types.intp,
      },
  )
  def rdist(x, y):
      """Reduced Euclidean distance.
  
      Parameters
      ----------
      x: array of shape (embedding_dim,)
      y: array of shape (embedding_dim,)
  
      Returns
      -------
      The squared euclidean distance between x and y
      """
      result = 0.0
      dim = x.shape[0]
      for i in range(dim):
          diff = x[i] - y[i]
          result += diff * diff
  
      return result
  
  
  def _optimize_layout_euclidean_single_epoch(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma,
      dim,
      move_other,
      alpha,
      epochs_per_negative_sample,
      epoch_of_next_negative_sample,
      epoch_of_next_sample,
      n,
      densmap_flag,
      dens_phi_sum,
      dens_re_sum,
      dens_re_cov,
      dens_re_std,
      dens_re_mean,
      dens_lambda,
      dens_R,
      dens_mu,
      dens_mu_tot,
  ):
      for i in numba.prange(epochs_per_sample.shape[0]):
          if epoch_of_next_sample[i] <= n:
              j = head[i]
              k = tail[i]
  
              current = head_embedding[j]
              other = tail_embedding[k]
  
              dist_squared = rdist(current, other)
  
              if densmap_flag:
                  phi = 1.0 / (1.0 + a * pow(dist_squared, b))
                  dphi_term = (
                      a * b * pow(dist_squared, b - 1) / (1.0 + a * pow(dist_squared, b))
                  )
  
                  q_jk = phi / dens_phi_sum[k]
                  q_kj = phi / dens_phi_sum[j]
  
                  drk = q_jk * (
                      (1.0 - b * (1 - phi)) / np.exp(dens_re_sum[k]) + dphi_term
                  )
                  drj = q_kj * (
                      (1.0 - b * (1 - phi)) / np.exp(dens_re_sum[j]) + dphi_term
                  )
  
                  re_std_sq = dens_re_std * dens_re_std
                  weight_k = (
                      dens_R[k]
                      - dens_re_cov * (dens_re_sum[k] - dens_re_mean) / re_std_sq
                  )
                  weight_j = (
                      dens_R[j]
                      - dens_re_cov * (dens_re_sum[j] - dens_re_mean) / re_std_sq
                  )
  
                  grad_cor_coeff = (
                      dens_lambda
                      * dens_mu_tot
                      * (weight_k * drk + weight_j * drj)
                      / (dens_mu[i] * dens_re_std)
                      / n_vertices
                  )
  
              if dist_squared > 0.0:
                  grad_coeff = -2.0 * a * b * pow(dist_squared, b - 1.0)
                  grad_coeff /= a * pow(dist_squared, b) + 1.0
              else:
                  grad_coeff = 0.0
  
              for d in range(dim):
                  grad_d = clip(grad_coeff * (current[d] - other[d]))
  
                  if densmap_flag:
                      grad_d += clip(2 * grad_cor_coeff * (current[d] - other[d]))
  
                  current[d] += grad_d * alpha
                  if move_other:
                      other[d] += -grad_d * alpha
  
              epoch_of_next_sample[i] += epochs_per_sample[i]
  
              n_neg_samples = int(
                  (n - epoch_of_next_negative_sample[i]) / epochs_per_negative_sample[i]
              )
  
              for p in range(n_neg_samples):
                  k = tau_rand_int(rng_state) % n_vertices
  
                  other = tail_embedding[k]
  
                  dist_squared = rdist(current, other)
  
                  if dist_squared > 0.0:
                      grad_coeff = 2.0 * gamma * b
                      grad_coeff /= (0.001 + dist_squared) * (
                          a * pow(dist_squared, b) + 1
                      )
                  elif j == k:
                      continue
                  else:
                      grad_coeff = 0.0
  
                  for d in range(dim):
                      if grad_coeff > 0.0:
                          grad_d = clip(grad_coeff * (current[d] - other[d]))
                      else:
                          grad_d = 4.0
                      current[d] += grad_d * alpha
  
              epoch_of_next_negative_sample[i] += (
                  n_neg_samples * epochs_per_negative_sample[i]
              )
  
  
  def _optimize_layout_euclidean_densmap_epoch_init(
      head_embedding,
      tail_embedding,
      head,
      tail,
      a,
      b,
      re_sum,
      phi_sum,
  ):
      re_sum.fill(0)
      phi_sum.fill(0)
  
      for i in numba.prange(head.size):
          j = head[i]
          k = tail[i]
  
          current = head_embedding[j]
          other = tail_embedding[k]
          dist_squared = rdist(current, other)
  
          phi = 1.0 / (1.0 + a * pow(dist_squared, b))
  
          re_sum[j] += phi * dist_squared
          re_sum[k] += phi * dist_squared
          phi_sum[j] += phi
          phi_sum[k] += phi
  
      epsilon = 1e-8
      for i in range(re_sum.size):
          re_sum[i] = np.log(epsilon + (re_sum[i] / phi_sum[i]))
  
  
  def optimize_layout_euclidean(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      parallel=False,
      verbose=False,
      densmap=False,
      densmap_kwds={},
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
      n_epochs: int
          The number of training epochs to use in optimization.
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
      epochs_per_samples: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
      a: float
          Parameter of differentiable approximation of right adjoint functor
      b: float
          Parameter of differentiable approximation of right adjoint functor
      rng_state: array of int64, shape (3,)
          The internal state of the rng
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
      parallel: bool (optional, default False)
          Whether to run the computation using numba parallel.
          Running in parallel is non-deterministic, and is not used
          if a random seed has been set, to ensure reproducibility.
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
      densmap: bool (optional, default False)
          Whether to use the density-augmented densMAP objective
      densmap_kwds: dict (optional, default {})
          Auxiliary data for densMAP
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      optimize_fn = numba.njit(
          _optimize_layout_euclidean_single_epoch, fastmath=True, parallel=parallel
      )
  
      if densmap:
          dens_init_fn = numba.njit(
              _optimize_layout_euclidean_densmap_epoch_init,
              fastmath=True,
              parallel=parallel,
          )
  
          dens_mu_tot = np.sum(densmap_kwds["mu_sum"]) / 2
          dens_lambda = densmap_kwds["lambda"]
          dens_R = densmap_kwds["R"]
          dens_mu = densmap_kwds["mu"]
          dens_phi_sum = np.zeros(n_vertices, dtype=np.float32)
          dens_re_sum = np.zeros(n_vertices, dtype=np.float32)
          dens_var_shift = densmap_kwds["var_shift"]
      else:
          dens_mu_tot = 0
          dens_lambda = 0
          dens_R = np.zeros(1, dtype=np.float32)
          dens_mu = np.zeros(1, dtype=np.float32)
          dens_phi_sum = np.zeros(1, dtype=np.float32)
          dens_re_sum = np.zeros(1, dtype=np.float32)
  
      for n in range(n_epochs):
  
          densmap_flag = (
              densmap
              and (densmap_kwds["lambda"] > 0)
              and (((n + 1) / float(n_epochs)) > (1 - densmap_kwds["frac"]))
          )
  
          if densmap_flag:
              dens_init_fn(
                  head_embedding,
                  tail_embedding,
                  head,
                  tail,
                  a,
                  b,
                  dens_re_sum,
                  dens_phi_sum,
              )
  
              dens_re_std = np.sqrt(np.var(dens_re_sum) + dens_var_shift)
              dens_re_mean = np.mean(dens_re_sum)
              dens_re_cov = np.dot(dens_re_sum, dens_R) / (n_vertices - 1)
          else:
              dens_re_std = 0
              dens_re_mean = 0
              dens_re_cov = 0
  
          optimize_fn(
              head_embedding,
              tail_embedding,
              head,
              tail,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              dim,
              move_other,
              alpha,
              epochs_per_negative_sample,
              epoch_of_next_negative_sample,
              epoch_of_next_sample,
              n,
              densmap_flag,
              dens_phi_sum,
              dens_re_sum,
              dens_re_cov,
              dens_re_std,
              dens_re_mean,
              dens_lambda,
              dens_R,
              dens_mu,
              dens_mu_tot,
          )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embedding
  
  
  @numba.njit(fastmath=True)
  def optimize_layout_generic(
      head_embedding,
      tail_embedding,
      head,
      tail,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      output_metric=dist.euclidean,
      output_metric_kwds=(),
      verbose=False,
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
  
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
  
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
  
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
  
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
  
      weight: array of shape (n_1_simplices)
          The membership weights of the 1-simplices.
  
      n_epochs: int
          The number of training epochs to use in optimization.
  
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
  
      epochs_per_sample: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      rng_state: array of int64, shape (3,)
          The internal state of the rng
  
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
  
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
  
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      for n in range(n_epochs):
          for i in range(epochs_per_sample.shape[0]):
              if epoch_of_next_sample[i] <= n:
                  j = head[i]
                  k = tail[i]
  
                  current = head_embedding[j]
                  other = tail_embedding[k]
  
                  dist_output, grad_dist_output = output_metric(
                      current, other, *output_metric_kwds
                  )
                  _, rev_grad_dist_output = output_metric(
                      other, current, *output_metric_kwds
                  )
  
                  if dist_output > 0.0:
                      w_l = pow((1 + a * pow(dist_output, 2 * b)), -1)
                  else:
                      w_l = 1.0
                  grad_coeff = 2 * b * (w_l - 1) / (dist_output + 1e-6)
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * grad_dist_output[d])
  
                      current[d] += grad_d * alpha
                      if move_other:
                          grad_d = clip(grad_coeff * rev_grad_dist_output[d])
                          other[d] += grad_d * alpha
  
                  epoch_of_next_sample[i] += epochs_per_sample[i]
  
                  n_neg_samples = int(
                      (n - epoch_of_next_negative_sample[i])
                      / epochs_per_negative_sample[i]
                  )
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % n_vertices
  
                      other = tail_embedding[k]
  
                      dist_output, grad_dist_output = output_metric(
                          current, other, *output_metric_kwds
                      )
  
                      if dist_output > 0.0:
                          w_l = pow((1 + a * pow(dist_output, 2 * b)), -1)
                      elif j == k:
                          continue
                      else:
                          w_l = 1.0
  
                      grad_coeff = gamma * 2 * b * w_l / (dist_output + 1e-6)
  
                      for d in range(dim):
                          grad_d = clip(grad_coeff * grad_dist_output[d])
                          current[d] += grad_d * alpha
  
                  epoch_of_next_negative_sample[i] += (
                      n_neg_samples * epochs_per_negative_sample[i]
                  )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embedding
  
  
  @numba.njit(fastmath=True)
  def optimize_layout_inverse(
      head_embedding,
      tail_embedding,
      head,
      tail,
      weight,
      sigmas,
      rhos,
      n_epochs,
      n_vertices,
      epochs_per_sample,
      a,
      b,
      rng_state,
      gamma=1.0,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      output_metric=dist.euclidean,
      output_metric_kwds=(),
      verbose=False,
      move_other=False,
  ):
      """Improve an embedding using stochastic gradient descent to minimize the
      fuzzy set cross entropy between the 1-skeletons of the high dimensional
      and low dimensional fuzzy simplicial sets. In practice this is done by
      sampling edges based on their membership strength (with the (1-p) terms
      coming from negative sampling similar to word2vec).
  
      Parameters
      ----------
      head_embedding: array of shape (n_samples, n_components)
          The initial embedding to be improved by SGD.
  
      tail_embedding: array of shape (source_samples, n_components)
          The reference embedding of embedded points. If not embedding new
          previously unseen points with respect to an existing embedding this
          is simply the head_embedding (again); otherwise it provides the
          existing embedding to embed with respect to.
  
      head: array of shape (n_1_simplices)
          The indices of the heads of 1-simplices with non-zero membership.
  
      tail: array of shape (n_1_simplices)
          The indices of the tails of 1-simplices with non-zero membership.
  
      weight: array of shape (n_1_simplices)
          The membership weights of the 1-simplices.
  
      n_epochs: int
          The number of training epochs to use in optimization.
  
      n_vertices: int
          The number of vertices (0-simplices) in the dataset.
  
      epochs_per_sample: array of shape (n_1_simplices)
          A float value of the number of epochs per 1-simplex. 1-simplices with
          weaker membership strength will have more epochs between being sampled.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      rng_state: array of int64, shape (3,)
          The internal state of the rng
  
      gamma: float (optional, default 1.0)
          Weight to apply to negative samples.
  
      initial_alpha: float (optional, default 1.0)
          Initial learning rate for the SGD.
  
      negative_sample_rate: int (optional, default 5)
          Number of negative samples to use per positive sample.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      move_other: bool (optional, default False)
          Whether to adjust tail_embedding alongside head_embedding
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized embedding.
      """
  
      dim = head_embedding.shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = epochs_per_sample / negative_sample_rate
      epoch_of_next_negative_sample = epochs_per_negative_sample.copy()
      epoch_of_next_sample = epochs_per_sample.copy()
  
      for n in range(n_epochs):
          for i in range(epochs_per_sample.shape[0]):
              if epoch_of_next_sample[i] <= n:
                  j = head[i]
                  k = tail[i]
  
                  current = head_embedding[j]
                  other = tail_embedding[k]
  
                  dist_output, grad_dist_output = output_metric(
                      current, other, *output_metric_kwds
                  )
  
                  w_l = weight[i]
                  grad_coeff = -(1 / (w_l * sigmas[k] + 1e-6))
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * grad_dist_output[d])
  
                      current[d] += grad_d * alpha
                      if move_other:
                          other[d] += -grad_d * alpha
  
                  epoch_of_next_sample[i] += epochs_per_sample[i]
  
                  n_neg_samples = int(
                      (n - epoch_of_next_negative_sample[i])
                      / epochs_per_negative_sample[i]
                  )
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % n_vertices
  
                      other = tail_embedding[k]
  
                      dist_output, grad_dist_output = output_metric(
                          current, other, *output_metric_kwds
                      )
  
                      # w_l = 0.0 # for negative samples, the edge does not exist
                      w_h = np.exp(-max(dist_output - rhos[k], 1e-6) / (sigmas[k] + 1e-6))
                      grad_coeff = -gamma * ((0 - w_h) / ((1 - w_h) * sigmas[k] + 1e-6))
  
                      for d in range(dim):
                          grad_d = clip(grad_coeff * grad_dist_output[d])
                          current[d] += grad_d * alpha
  
                  epoch_of_next_negative_sample[i] += (
                      n_neg_samples * epochs_per_negative_sample[i]
                  )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embedding
  
  
  def _optimize_layout_aligned_euclidean_single_epoch(
      head_embeddings,
      tail_embeddings,
      heads,
      tails,
      epochs_per_sample,
      a,
      b,
      regularisation_weights,
      relations,
      rng_state,
      gamma,
      lambda_,
      dim,
      move_other,
      alpha,
      epochs_per_negative_sample,
      epoch_of_next_negative_sample,
      epoch_of_next_sample,
      n,
  ):
      n_embeddings = len(heads)
      window_size = (relations.shape[1] - 1) // 2
  
      max_n_edges = 0
      for e_p_s in epochs_per_sample:
          if e_p_s.shape[0] >= max_n_edges:
              max_n_edges = e_p_s.shape[0]
  
      embedding_order = np.arange(n_embeddings).astype(np.int32)
      np.random.seed(abs(rng_state[0]))
      np.random.shuffle(embedding_order)
  
      for i in range(max_n_edges):
          for m in embedding_order:
              if i < epoch_of_next_sample[m].shape[0] and epoch_of_next_sample[m][i] <= n:
                  j = heads[m][i]
                  k = tails[m][i]
  
                  current = head_embeddings[m][j]
                  other = tail_embeddings[m][k]
  
                  dist_squared = rdist(current, other)
  
                  if dist_squared > 0.0:
                      grad_coeff = -2.0 * a * b * pow(dist_squared, b - 1.0)
                      grad_coeff /= a * pow(dist_squared, b) + 1.0
                  else:
                      grad_coeff = 0.0
  
                  for d in range(dim):
                      grad_d = clip(grad_coeff * (current[d] - other[d]))
  
                      for offset in range(-window_size, window_size):
                          neighbor_m = m + offset
                          if (
                              neighbor_m >= 0
                              and neighbor_m < n_embeddings
                              and offset != 0
                          ):
                              identified_index = relations[m, offset + window_size, j]
                              if identified_index >= 0:
                                  grad_d -= clip(
                                      (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                      * regularisation_weights[m, offset + window_size, j]
                                      * (
                                          current[d]
                                          - head_embeddings[neighbor_m][
                                              identified_index, d
                                          ]
                                      )
                                  )
  
                      current[d] += clip(grad_d) * alpha
                      if move_other:
                          other_grad_d = clip(grad_coeff * (other[d] - current[d]))
  
                          for offset in range(-window_size, window_size):
                              neighbor_m = m + offset
                              if (
                                  neighbor_m >= 0
                                  and neighbor_m < n_embeddings
                                  and offset != 0
                              ):
                                  identified_index = relations[m, offset + window_size, k]
                                  if identified_index >= 0:
                                      grad_d -= clip(
                                          (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                          * regularisation_weights[
                                              m, offset + window_size, k
                                          ]
                                          * (
                                              other[d]
                                              - head_embeddings[neighbor_m][
                                                  identified_index, d
                                              ]
                                          )
                                      )
  
                          other[d] += clip(other_grad_d) * alpha
  
                  epoch_of_next_sample[m][i] += epochs_per_sample[m][i]
  
                  if epochs_per_negative_sample[m][i] > 0:
                      n_neg_samples = int(
                          (n - epoch_of_next_negative_sample[m][i])
                          / epochs_per_negative_sample[m][i]
                      )
                  else:
                      n_neg_samples = 0
  
                  for p in range(n_neg_samples):
                      k = tau_rand_int(rng_state) % tail_embeddings[m].shape[0]
  
                      other = tail_embeddings[m][k]
  
                      dist_squared = rdist(current, other)
  
                      if dist_squared > 0.0:
                          grad_coeff = 2.0 * gamma * b
                          grad_coeff /= (0.001 + dist_squared) * (
                              a * pow(dist_squared, b) + 1
                          )
                      elif j == k:
                          continue
                      else:
                          grad_coeff = 0.0
  
                      for d in range(dim):
                          if grad_coeff > 0.0:
                              grad_d = clip(grad_coeff * (current[d] - other[d]))
                          else:
                              grad_d = 4.0
  
                          for offset in range(-window_size, window_size):
                              neighbor_m = m + offset
                              if (
                                  neighbor_m >= 0
                                  and neighbor_m < n_embeddings
                                  and offset != 0
                              ):
                                  identified_index = relations[m, offset + window_size, j]
                                  if identified_index >= 0:
                                      grad_d -= clip(
                                          (lambda_ * np.exp(-(np.abs(offset) - 1)))
                                          * regularisation_weights[
                                              m, offset + window_size, j
                                          ]
                                          * (
                                              current[d]
                                              - head_embeddings[neighbor_m][
                                                  identified_index, d
                                              ]
                                          )
                                      )
  
                          current[d] += clip(grad_d) * alpha
  
                  epoch_of_next_negative_sample[m][i] += (
                      n_neg_samples * epochs_per_negative_sample[m][i]
                  )
  
  
  def optimize_layout_aligned_euclidean(
      head_embeddings,
      tail_embeddings,
      heads,
      tails,
      n_epochs,
      epochs_per_sample,
      regularisation_weights,
      relations,
      rng_state,
      a=1.576943460405378,
      b=0.8950608781227859,
      gamma=1.0,
      lambda_=5e-3,
      initial_alpha=1.0,
      negative_sample_rate=5.0,
      parallel=True,
      verbose=False,
      move_other=False,
  ):
      dim = head_embeddings[0].shape[1]
      alpha = initial_alpha
  
      epochs_per_negative_sample = numba.typed.List.empty_list(numba.types.float32[::1])
      epoch_of_next_negative_sample = numba.typed.List.empty_list(
          numba.types.float32[::1]
      )
      epoch_of_next_sample = numba.typed.List.empty_list(numba.types.float32[::1])
  
      for m in range(len(heads)):
          epochs_per_negative_sample.append(
              epochs_per_sample[m].astype(np.float32) / negative_sample_rate
          )
          epoch_of_next_negative_sample.append(
              epochs_per_negative_sample[m].astype(np.float32)
          )
          epoch_of_next_sample.append(epochs_per_sample[m].astype(np.float32))
  
      optimize_fn = numba.njit(
          _optimize_layout_aligned_euclidean_single_epoch,
          fastmath=True,
          parallel=parallel,
      )
  
      for n in range(n_epochs):
          optimize_fn(
              head_embeddings,
              tail_embeddings,
              heads,
              tails,
              epochs_per_sample,
              a,
              b,
              regularisation_weights,
              relations,
              rng_state,
              gamma,
              lambda_,
              dim,
              move_other,
              alpha,
              epochs_per_negative_sample,
              epoch_of_next_negative_sample,
              epoch_of_next_sample,
              n,
          )
  
          alpha = initial_alpha * (1.0 - (float(n) / float(n_epochs)))
  
          if verbose and n % int(n_epochs / 10) == 0:
              print("\tcompleted ", n, " / ", n_epochs, "epochs")
  
      return head_embeddings
  
files: umap/layouts.py
files: # Author: Leland McInnes <leland.mcinnes@gmail.com>
  #
  # License: BSD 3 clause
  from __future__ import print_function
  
  import locale
  from warnings import warn
  import time
  
  from scipy.optimize import curve_fit
  from sklearn.base import BaseEstimator
  from sklearn.utils import check_random_state, check_array
  from sklearn.utils.validation import check_is_fitted
  from sklearn.metrics import pairwise_distances
  from sklearn.preprocessing import normalize
  from sklearn.neighbors import KDTree
  
  try:
      import joblib
  except ImportError:
      # sklearn.externals.joblib is deprecated in 0.21, will be removed in 0.23
      from sklearn.externals import joblib
  
  import numpy as np
  import scipy.sparse
  from scipy.sparse import tril as sparse_tril, triu as sparse_triu
  import scipy.sparse.csgraph
  import numba
  
  import umap.distances as dist
  
  import umap.sparse as sparse
  
  from umap.utils import (
      submatrix,
      ts,
      csr_unique,
      fast_knn_indices,
  )
  from umap.spectral import spectral_layout
  from umap.layouts import (
      optimize_layout_euclidean,
      optimize_layout_generic,
      optimize_layout_inverse,
  )
  
  from pynndescent import NNDescent
  from pynndescent.distances import named_distances as pynn_named_distances
  from pynndescent.sparse import sparse_named_distances as pynn_sparse_named_distances
  
  locale.setlocale(locale.LC_NUMERIC, "C")
  
  INT32_MIN = np.iinfo(np.int32).min + 1
  INT32_MAX = np.iinfo(np.int32).max - 1
  
  SMOOTH_K_TOLERANCE = 1e-5
  MIN_K_DIST_SCALE = 1e-3
  NPY_INFINITY = np.inf
  
  DISCONNECTION_DISTANCES = {
      "correlation": 2,
      "cosine": 2,
      "hellinger": 1,
      "jaccard": 1,
      "dice": 1,
  }
  
  
  def flatten_iter(container):
      for i in container:
          if isinstance(i, (list, tuple)):
              for j in flatten_iter(i):
                  yield j
          else:
              yield i
  
  
  def flattened(container):
      return tuple(flatten_iter(container))
  
  
  def breadth_first_search(adjmat, start, min_vertices):
      explored = []
      queue = [start]
      levels = {}
      levels[start] = 0
      max_level = np.inf
      visited = [start]
  
      while queue:
          node = queue.pop(0)
          explored.append(node)
          if max_level == np.inf and len(explored) > min_vertices:
              max_level = max(levels.values())
  
          if levels[node] + 1 < max_level:
              neighbors = adjmat[node].indices
              for neighbour in neighbors:
                  if neighbour not in visited:
                      queue.append(neighbour)
                      visited.append(neighbour)
  
                      levels[neighbour] = levels[node] + 1
  
      return np.array(explored)
  
  
  def raise_disconnected_warning(
      edges_removed,
      vertices_disconnected,
      disconnection_distance,
      total_rows,
      threshold=0.1,
      verbose=False,
  ):
      """A simple wrapper function to avoid large amounts of code repetition."""
      if verbose & (vertices_disconnected == 0) & (edges_removed > 0):
          print(
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.  "
              f"This is not a problem as no vertices were disconnected."
          )
      elif (vertices_disconnected > 0) & (
          vertices_disconnected <= threshold * total_rows
      ):
          warn(
              f"A few of your vertices were disconnected from the manifold.  This shouldn't cause problems.\n"
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.\n"
              f"It has only fully disconnected {vertices_disconnected} vertices.\n"
              f"Use umap.utils.disconnected_vertices() to identify them.",
          )
      elif vertices_disconnected > threshold * total_rows:
          warn(
              f"A large number of your vertices were disconnected from the manifold.\n"
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.\n"
              f"It has fully disconnected {vertices_disconnected} vertices.\n"
              f"You might consider using find_disconnected_points() to find and remove these points from your data.\n"
              f"Use umap.utils.disconnected_vertices() to identify them.",
          )
  
  
  @numba.njit(
      locals={
          "psum": numba.types.float32,
          "lo": numba.types.float32,
          "mid": numba.types.float32,
          "hi": numba.types.float32,
      },
      fastmath=True,
  )  # benchmarking `parallel=True` shows it to *decrease* performance
  def smooth_knn_dist(distances, k, n_iter=64, local_connectivity=1.0, bandwidth=1.0):
      """Compute a continuous version of the distance to the kth nearest
      neighbor. That is, this is similar to knn-distance but allows continuous
      k values rather than requiring an integral k. In essence we are simply
      computing the distance such that the cardinality of fuzzy set we generate
      is k.
  
      Parameters
      ----------
      distances: array of shape (n_samples, n_neighbors)
          Distances to nearest neighbors for each samples. Each row should be a
          sorted list of distances to a given samples nearest neighbors.
  
      k: float
          The number of nearest neighbors to approximate for.
  
      n_iter: int (optional, default 64)
          We need to binary search for the correct distance value. This is the
          max number of iterations to use in such a search.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      bandwidth: float (optional, default 1)
          The target bandwidth of the kernel, larger values will produce
          larger return values.
  
      Returns
      -------
      knn_dist: array of shape (n_samples,)
          The distance to kth nearest neighbor, as suitably approximated.
  
      nn_dist: array of shape (n_samples,)
          The distance to the 1st nearest neighbor for each point.
      """
      target = np.log2(k) * bandwidth
      rho = np.zeros(distances.shape[0], dtype=np.float32)
      result = np.zeros(distances.shape[0], dtype=np.float32)
  
      mean_distances = np.mean(distances)
  
      for i in range(distances.shape[0]):
          lo = 0.0
          hi = NPY_INFINITY
          mid = 1.0
  
          # TODO: This is very inefficient, but will do for now. FIXME
          ith_distances = distances[i]
          non_zero_dists = ith_distances[ith_distances > 0.0]
          if non_zero_dists.shape[0] >= local_connectivity:
              index = int(np.floor(local_connectivity))
              interpolation = local_connectivity - index
              if index > 0:
                  rho[i] = non_zero_dists[index - 1]
                  if interpolation > SMOOTH_K_TOLERANCE:
                      rho[i] += interpolation * (
                          non_zero_dists[index] - non_zero_dists[index - 1]
                      )
              else:
                  rho[i] = interpolation * non_zero_dists[0]
          elif non_zero_dists.shape[0] > 0:
              rho[i] = np.max(non_zero_dists)
  
          for n in range(n_iter):
  
              psum = 0.0
              for j in range(1, distances.shape[1]):
                  d = distances[i, j] - rho[i]
                  if d > 0:
                      psum += np.exp(-(d / mid))
                  else:
                      psum += 1.0
  
              if np.fabs(psum - target) < SMOOTH_K_TOLERANCE:
                  break
  
              if psum > target:
                  hi = mid
                  mid = (lo + hi) / 2.0
              else:
                  lo = mid
                  if hi == NPY_INFINITY:
                      mid *= 2
                  else:
                      mid = (lo + hi) / 2.0
  
          result[i] = mid
  
          # TODO: This is very inefficient, but will do for now. FIXME
          if rho[i] > 0.0:
              mean_ith_distances = np.mean(ith_distances)
              if result[i] < MIN_K_DIST_SCALE * mean_ith_distances:
                  result[i] = MIN_K_DIST_SCALE * mean_ith_distances
          else:
              if result[i] < MIN_K_DIST_SCALE * mean_distances:
                  result[i] = MIN_K_DIST_SCALE * mean_distances
  
      return result, rho
  
  
  def nearest_neighbors(
      X,
      n_neighbors,
      metric,
      metric_kwds,
      angular,
      random_state,
      low_memory=True,
      use_pynndescent=True,
      n_jobs=-1,
      verbose=False,
  ):
      """Compute the ``n_neighbors`` nearest points for each data point in ``X``
      under ``metric``. This may be exact, but more likely is approximated via
      nearest neighbor descent.
  
      Parameters
      ----------
      X: array of shape (n_samples, n_features)
          The input data to compute the k-neighbor graph of.
  
      n_neighbors: int
          The number of nearest neighbors to compute for each sample in ``X``.
  
      metric: string or callable
          The metric to use for the computation.
  
      metric_kwds: dict
          Any arguments to pass to the metric computation function.
  
      angular: bool
          Whether to use angular rp trees in NN approximation.
  
      random_state: np.random state
          The random state to use for approximate NN computations.
  
      low_memory: bool (optional, default True)
          Whether to pursue lower memory NNdescent.
  
      verbose: bool (optional, default False)
          Whether to print status data during the computation.
  
      Returns
      -------
      knn_indices: array of shape (n_samples, n_neighbors)
          The indices on the ``n_neighbors`` closest points in the dataset.
  
      knn_dists: array of shape (n_samples, n_neighbors)
          The distances to the ``n_neighbors`` closest points in the dataset.
  
      rp_forest: list of trees
          The random projection forest used for searching (if used, None otherwise)
      """
      if verbose:
          print(ts(), "Finding Nearest Neighbors")
  
      if metric == "precomputed":
          # Note that this does not support sparse distance matrices yet ...
          # Compute indices of n nearest neighbors
          knn_indices = fast_knn_indices(X, n_neighbors)
          # knn_indices = np.argsort(X)[:, :n_neighbors]
          # Compute the nearest neighbor distances
          #   (equivalent to np.sort(X)[:,:n_neighbors])
          knn_dists = X[np.arange(X.shape[0])[:, None], knn_indices].copy()
          # Prune any nearest neighbours that are infinite distance apart.
          disconnected_index = knn_dists == np.inf
          knn_indices[disconnected_index] = -1
  
          knn_search_index = None
      else:
          # TODO: Hacked values for now
          n_trees = min(64, 5 + int(round((X.shape[0]) ** 0.5 / 20.0)))
          n_iters = max(5, int(round(np.log2(X.shape[0]))))
  
          knn_search_index = NNDescent(
              X,
              n_neighbors=n_neighbors,
              metric=metric,
              metric_kwds=metric_kwds,
              random_state=random_state,
              n_trees=n_trees,
              n_iters=n_iters,
              max_candidates=60,
              low_memory=low_memory,
              n_jobs=n_jobs,
              verbose=verbose,
          )
          knn_indices, knn_dists = knn_search_index.neighbor_graph
  
      if verbose:
          print(ts(), "Finished Nearest Neighbor Search")
      return knn_indices, knn_dists, knn_search_index
  
  
  @numba.njit(
      locals={
          "knn_dists": numba.types.float32[:, ::1],
          "sigmas": numba.types.float32[::1],
          "rhos": numba.types.float32[::1],
          "val": numba.types.float32,
      },
      parallel=True,
      fastmath=True,
  )
  def compute_membership_strengths(
      knn_indices,
      knn_dists,
      sigmas,
      rhos,
      return_dists=False,
      bipartite=False,
  ):
      """Construct the membership strength data for the 1-skeleton of each local
      fuzzy simplicial set -- this is formed as a sparse matrix where each row is
      a local fuzzy simplicial set, with a membership strength for the
      1-simplex to each other data point.
  
      Parameters
      ----------
      knn_indices: array of shape (n_samples, n_neighbors)
          The indices on the ``n_neighbors`` closest points in the dataset.
  
      knn_dists: array of shape (n_samples, n_neighbors)
          The distances to the ``n_neighbors`` closest points in the dataset.
  
      sigmas: array of shape(n_samples)
          The normalization factor derived from the metric tensor approximation.
  
      rhos: array of shape(n_samples)
          The local connectivity adjustment.
  
      return_dists: bool (optional, default False)
          Whether to return the pairwise distance associated with each edge
  
      bipartite: bool (optional, default False)
          Does the nearest neighbour set represent a bipartite graph?  That is are the
          nearest neighbour indices from the same point set as the row indices?
  
      Returns
      -------
      rows: array of shape (n_samples * n_neighbors)
          Row data for the resulting sparse matrix (coo format)
  
      cols: array of shape (n_samples * n_neighbors)
          Column data for the resulting sparse matrix (coo format)
  
      vals: array of shape (n_samples * n_neighbors)
          Entries for the resulting sparse matrix (coo format)
  
      dists: array of shape (n_samples * n_neighbors)
          Distance associated with each entry in the resulting sparse matrix
      """
      n_samples = knn_indices.shape[0]
      n_neighbors = knn_indices.shape[1]
  
      rows = np.zeros(knn_indices.size, dtype=np.int32)
      cols = np.zeros(knn_indices.size, dtype=np.int32)
      vals = np.zeros(knn_indices.size, dtype=np.float32)
      if return_dists:
          dists = np.zeros(knn_indices.size, dtype=np.float32)
      else:
          dists = None
  
      for i in range(n_samples):
          for j in range(n_neighbors):
              if knn_indices[i, j] == -1:
                  continue  # We didn't get the full knn for i
              # If applied to an adjacency matrix points shouldn't be similar to themselves.
              # If applied to an incidence matrix (or bipartite) then the row and column indices are different.
              if (bipartite == False) & (knn_indices[i, j] == i):
                  val = 0.0
              elif knn_dists[i, j] - rhos[i] <= 0.0 or sigmas[i] == 0.0:
                  val = 1.0
              else:
                  val = np.exp(-((knn_dists[i, j] - rhos[i]) / (sigmas[i])))
  
              rows[i * n_neighbors + j] = i
              cols[i * n_neighbors + j] = knn_indices[i, j]
              vals[i * n_neighbors + j] = val
              if return_dists:
                  dists[i * n_neighbors + j] = knn_dists[i, j]
  
      return rows, cols, vals, dists
  
  
  def fuzzy_simplicial_set(
      X,
      n_neighbors,
      random_state,
      metric,
      metric_kwds={},
      knn_indices=None,
      knn_dists=None,
      angular=False,
      set_op_mix_ratio=1.0,
      local_connectivity=1.0,
      apply_set_operations=True,
      verbose=False,
      return_dists=None,
  ):
      """Given a set of data X, a neighborhood size, and a measure of distance
      compute the fuzzy simplicial set (here represented as a fuzzy graph in
      the form of a sparse matrix) associated to the data. This is done by
      locally approximating geodesic distance at each point, creating a fuzzy
      simplicial set for each such point, and then combining all the local
      fuzzy simplicial sets into a global one via a fuzzy union.
  
      Parameters
      ----------
      X: array of shape (n_samples, n_features)
          The data to be modelled as a fuzzy simplicial set.
  
      n_neighbors: int
          The number of neighbors to use to approximate geodesic distance.
          Larger numbers induce more global estimates of the manifold that can
          miss finer detail, while smaller values will focus on fine manifold
          structure to the detriment of the larger picture.
  
      random_state: numpy RandomState or equivalent
          A state capable being used as a numpy random state.
  
      metric: string or function (optional, default 'euclidean')
          The metric to use to compute distances in high dimensional space.
          If a string is passed it must match a valid predefined metric. If
          a general metric is required a function that takes two 1d arrays and
          returns a float can be provided. For performance purposes it is
          required that this be a numba jit'd function. Valid string metrics
          include:
              * euclidean (or l2)
              * manhattan (or l1)
              * cityblock
              * braycurtis
              * canberra
              * chebyshev
              * correlation
              * cosine
              * dice
              * hamming
              * jaccard
              * kulsinski
              * ll_dirichlet
              * mahalanobis
              * matching
              * minkowski
              * rogerstanimoto
              * russellrao
              * seuclidean
              * sokalmichener
              * sokalsneath
              * sqeuclidean
              * yule
              * wminkowski
  
          Metrics that take arguments (such as minkowski, mahalanobis etc.)
          can have arguments passed via the metric_kwds dictionary. At this
          time care must be taken and dictionary elements must be ordered
          appropriately; this will hopefully be fixed in the future.
  
      metric_kwds: dict (optional, default {})
          Arguments to pass on to the metric, such as the ``p`` value for
          Minkowski distance.
  
      knn_indices: array of shape (n_samples, n_neighbors) (optional)
          If the k-nearest neighbors of each point has already been calculated
          you can pass them in here to save computation time. This should be
          an array with the indices of the k-nearest neighbors as a row for
          each data point.
  
      knn_dists: array of shape (n_samples, n_neighbors) (optional)
          If the k-nearest neighbors of each point has already been calculated
          you can pass them in here to save computation time. This should be
          an array with the distances of the k-nearest neighbors as a row for
          each data point.
  
      angular: bool (optional, default False)
          Whether to use angular/cosine distance for the random projection
          forest for seeding NN-descent to determine approximate nearest
          neighbors.
  
      set_op_mix_ratio: float (optional, default 1.0)
          Interpolate between (fuzzy) union and intersection as the set operation
          used to combine local fuzzy simplicial sets to obtain a global fuzzy
          simplicial sets. Both fuzzy set operations use the product t-norm.
          The value of this parameter should be between 0.0 and 1.0; a value of
          1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy
          intersection.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      return_dists: bool or None (optional, default None)
          Whether to return the pairwise distance associated with each edge.
  
      Returns
      -------
      fuzzy_simplicial_set: coo_matrix
          A fuzzy simplicial set represented as a sparse matrix. The (i,
          j) entry of the matrix represents the membership strength of the
          1-simplex between the ith and jth sample points.
      """
      if knn_indices is None or knn_dists is None:
          knn_indices, knn_dists, _ = nearest_neighbors(
              X,
              n_neighbors,
              metric,
              metric_kwds,
              angular,
              random_state,
              verbose=verbose,
          )
  
      knn_dists = knn_dists.astype(np.float32)
  
      sigmas, rhos = smooth_knn_dist(
          knn_dists,
          float(n_neighbors),
          local_connectivity=float(local_connectivity),
      )
  
      rows, cols, vals, dists = compute_membership_strengths(
          knn_indices, knn_dists, sigmas, rhos, return_dists
      )
  
      result = scipy.sparse.coo_matrix(
          (vals, (rows, cols)), shape=(X.shape[0], X.shape[0])
      )
      result.eliminate_zeros()
  
      if apply_set_operations:
          transpose = result.transpose()
  
          prod_matrix = result.multiply(transpose)
  
          result = (
              set_op_mix_ratio * (result + transpose - prod_matrix)
              + (1.0 - set_op_mix_ratio) * prod_matrix
          )
  
      result.eliminate_zeros()
  
      if return_dists is None:
          return result, sigmas, rhos
      else:
          if return_dists:
              dmat = scipy.sparse.coo_matrix(
                  (dists, (rows, cols)), shape=(X.shape[0], X.shape[0])
              )
  
              dists = dmat.maximum(dmat.transpose()).todok()
          else:
              dists = None
  
          return result, sigmas, rhos, dists
  
  
  @numba.njit()
  def fast_intersection(rows, cols, values, target, unknown_dist=1.0, far_dist=5.0):
      """Under the assumption of categorical distance for the intersecting
      simplicial set perform a fast intersection.
  
      Parameters
      ----------
      rows: array
          An array of the row of each non-zero in the sparse matrix
          representation.
  
      cols: array
          An array of the column of each non-zero in the sparse matrix
          representation.
  
      values: array
          An array of the value of each non-zero in the sparse matrix
          representation.
  
      target: array of shape (n_samples)
          The categorical labels to use in the intersection.
  
      unknown_dist: float (optional, default 1.0)
          The distance an unknown label (-1) is assumed to be from any point.
  
      far_dist float (optional, default 5.0)
          The distance between unmatched labels.
  
      Returns
      -------
      None
      """
      for nz in range(rows.shape[0]):
          i = rows[nz]
          j = cols[nz]
          if (target[i] == -1) or (target[j] == -1):
              values[nz] *= np.exp(-unknown_dist)
          elif target[i] != target[j]:
              values[nz] *= np.exp(-far_dist)
  
      return
  
  
  @numba.jit()
  def fast_metric_intersection(
      rows, cols, values, discrete_space, metric, metric_args, scale
  ):
      """Under the assumption of categorical distance for the intersecting
      simplicial set perform a fast intersection.
  
      Parameters
      ----------
      rows: array
          An array of the row of each non-zero in the sparse matrix
          representation.
  
      cols: array
          An array of the column of each non-zero in the sparse matrix
          representation.
  
      values: array of shape
          An array of the values of each non-zero in the sparse matrix
          representation.
  
      discrete_space: array of shape (n_samples, n_features)
          The vectors of categorical labels to use in the intersection.
  
      metric: numba function
          The function used to calculate distance over the target array.
  
      scale: float
          A scaling to apply to the metric.
  
      Returns
      -------
      None
      """
      for nz in range(rows.shape[0]):
          i = rows[nz]
          j = cols[nz]
          dist = metric(discrete_space[i], discrete_space[j], *metric_args)
          values[nz] *= np.exp(-(scale * dist))
  
      return
  
  
  @numba.njit()
  def reprocess_row(probabilities, k=15, n_iters=32):
      target = np.log2(k)
  
      lo = 0.0
      hi = NPY_INFINITY
      mid = 1.0
  
      for n in range(n_iters):
  
          psum = 0.0
          for j in range(probabilities.shape[0]):
              psum += pow(probabilities[j], mid)
  
          if np.fabs(psum - target) < SMOOTH_K_TOLERANCE:
              break
  
          if psum < target:
              hi = mid
              mid = (lo + hi) / 2.0
          else:
              lo = mid
              if hi == NPY_INFINITY:
                  mid *= 2
              else:
                  mid = (lo + hi) / 2.0
  
      return np.power(probabilities, mid)
  
  
  @numba.njit()
  def reset_local_metrics(simplicial_set_indptr, simplicial_set_data):
      for i in range(simplicial_set_indptr.shape[0] - 1):
          simplicial_set_data[
              simplicial_set_indptr[i] : simplicial_set_indptr[i + 1]
          ] = reprocess_row(
              simplicial_set_data[simplicial_set_indptr[i] : simplicial_set_indptr[i + 1]]
          )
      return
  
  
  def reset_local_connectivity(simplicial_set, reset_local_metric=False):
      """Reset the local connectivity requirement -- each data sample should
      have complete confidence in at least one 1-simplex in the simplicial set.
      We can enforce this by locally rescaling confidences, and then remerging the
      different local simplicial sets together.
  
      Parameters
      ----------
      simplicial_set: sparse matrix
          The simplicial set for which to recalculate with respect to local
          connectivity.
  
      Returns
      -------
      simplicial_set: sparse_matrix
          The recalculated simplicial set, now with the local connectivity
          assumption restored.
      """
      simplicial_set = normalize(simplicial_set, norm="max")
      if reset_local_metric:
          simplicial_set = simplicial_set.tocsr()
          reset_local_metrics(simplicial_set.indptr, simplicial_set.data)
          simplicial_set = simplicial_set.tocoo()
      transpose = simplicial_set.transpose()
      prod_matrix = simplicial_set.multiply(transpose)
      simplicial_set = simplicial_set + transpose - prod_matrix
      simplicial_set.eliminate_zeros()
  
      return simplicial_set
  
  
  def discrete_metric_simplicial_set_intersection(
      simplicial_set,
      discrete_space,
      unknown_dist=1.0,
      far_dist=5.0,
      metric=None,
      metric_kws={},
      metric_scale=1.0,
  ):
      """Combine a fuzzy simplicial set with another fuzzy simplicial set
      generated from discrete metric data using discrete distances. The target
      data is assumed to be categorical label data (a vector of labels),
      and this will update the fuzzy simplicial set to respect that label data.
  
      TODO: optional category cardinality based weighting of distance
  
      Parameters
      ----------
      simplicial_set: sparse matrix
          The input fuzzy simplicial set.
  
      discrete_space: array of shape (n_samples)
          The categorical labels to use in the intersection.
  
      unknown_dist: float (optional, default 1.0)
          The distance an unknown label (-1) is assumed to be from any point.
  
      far_dist: float (optional, default 5.0)
          The distance between unmatched labels.
  
      metric: str (optional, default None)
          If not None, then use this metric to determine the
          distance between values.
  
      metric_scale: float (optional, default 1.0)
          If using a custom metric scale the distance values by
          this value -- this controls the weighting of the
          intersection. Larger values weight more toward target.
  
      Returns
      -------
      simplicial_set: sparse matrix
          The resulting intersected fuzzy simplicial set.
      """
      simplicial_set = simplicial_set.tocoo()
  
      if metric is not None:
          # We presume target is now a 2d array, with each row being a
          # vector of target info
          if metric in dist.named_distances:
              metric_func = dist.named_distances[metric]
          else:
              raise ValueError("Discrete intersection metric is not recognized")
  
          fast_metric_intersection(
              simplicial_set.row,
              simplicial_set.col,
              simplicial_set.data,
              discrete_space,
              metric_func,
              tuple(metric_kws.values()),
              metric_scale,
          )
      else:
          fast_intersection(
              simplicial_set.row,
              simplicial_set.col,
              simplicial_set.data,
              discrete_space,
              unknown_dist,
              far_dist,
          )
  
      simplicial_set.eliminate_zeros()
  
      return reset_local_connectivity(simplicial_set)
  
  
  def general_simplicial_set_intersection(
      simplicial_set1, simplicial_set2, weight=0.5, right_complement=False
  ):
  
      if right_complement:
          result = simplicial_set1.tocoo()
      else:
          result = (simplicial_set1 + simplicial_set2).tocoo()
      left = simplicial_set1.tocsr()
      right = simplicial_set2.tocsr()
  
      sparse.general_sset_intersection(
          left.indptr,
          left.indices,
          left.data,
          right.indptr,
          right.indices,
          right.data,
          result.row,
          result.col,
          result.data,
          mix_weight=weight,
          right_complement=right_complement,
      )
  
      return result
  
  
  def general_simplicial_set_union(simplicial_set1, simplicial_set2):
      result = (simplicial_set1 + simplicial_set2).tocoo()
      left = simplicial_set1.tocsr()
      right = simplicial_set2.tocsr()
  
      sparse.general_sset_union(
          left.indptr,
          left.indices,
          left.data,
          right.indptr,
          right.indices,
          right.data,
          result.row,
          result.col,
          result.data,
      )
  
      return result
  
  
  def make_epochs_per_sample(weights, n_epochs):
      """Given a set of weights and number of epochs generate the number of
      epochs per sample for each weight.
  
      Parameters
      ----------
      weights: array of shape (n_1_simplices)
          The weights ofhow much we wish to sample each 1-simplex.
  
      n_epochs: int
          The total number of epochs we want to train for.
  
      Returns
      -------
      An array of number of epochs per sample, one for each 1-simplex.
      """
      result = -1.0 * np.ones(weights.shape[0], dtype=np.float64)
      n_samples = n_epochs * (weights / weights.max())
      result[n_samples > 0] = float(n_epochs) / n_samples[n_samples > 0]
      return result
  
  
  def simplicial_set_embedding(
      data,
      graph,
      n_components,
      initial_alpha,
      a,
      b,
      gamma,
      negative_sample_rate,
      n_epochs,
      init,
      random_state,
      metric,
      metric_kwds,
      densmap,
      densmap_kwds,
      output_dens,
      output_metric=dist.named_distances_with_gradients["euclidean"],
      output_metric_kwds={},
      euclidean_output=True,
      parallel=False,
      verbose=False,
  ):
      """Perform a fuzzy simplicial set embedding, using a specified
      initialisation method and then minimizing the fuzzy set cross entropy
      between the 1-skeletons of the high and low dimensional fuzzy simplicial
      sets.
  
      Parameters
      ----------
      data: array of shape (n_samples, n_features)
          The source data to be embedded by UMAP.
  
      graph: sparse matrix
          The 1-skeleton of the high dimensional fuzzy simplicial set as
          represented by a graph for which we require a sparse matrix for the
          (weighted) adjacency matrix.
  
      n_components: int
          The dimensionality of the euclidean space into which to embed the data.
  
      initial_alpha: float
          Initial learning rate for the SGD.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      gamma: float
          Weight to apply to negative samples.
  
      negative_sample_rate: int (optional, default 5)
          The number of negative samples to select per positive sample
          in the optimization process. Increasing this value will result
          in greater repulsive force being applied, greater optimization
          cost, but slightly more accuracy.
  
      n_epochs: int (optional, default 0), or list of int
          The number of training epochs to be used in optimizing the
          low dimensional embedding. Larger values result in more accurate
          embeddings. If 0 is specified a value will be selected based on
          the size of the input dataset (200 for large datasets, 500 for small).
          If a list of int is specified, then the intermediate embeddings at the
          different epochs specified in that list are returned in
          ``aux_data["embedding_list"]``.
  
      init: string
          How to initialize the low dimensional embedding. Options are:
              * 'spectral': use a spectral embedding of the fuzzy 1-skeleton
              * 'random': assign initial embedding positions at random.
              * A numpy array of initial embedding positions.
  
      random_state: numpy RandomState or equivalent
          A state capable being used as a numpy random state.
  
      metric: string or callable
          The metric used to measure distance in high dimensional space; used if
          multiple connected components need to be layed out.
  
      metric_kwds: dict
          Key word arguments to be passed to the metric function; used if
          multiple connected components need to be layed out.
  
      densmap: bool
          Whether to use the density-augmented objective function to optimize
          the embedding according to the densMAP algorithm.
  
      densmap_kwds: dict
          Key word arguments to be used by the densMAP optimization.
  
      output_dens: bool
          Whether to output local radii in the original data and the embedding.
  
      output_metric: function
          Function returning the distance between two points in embedding space and
          the gradient of the distance wrt the first argument.
  
      output_metric_kwds: dict
          Key word arguments to be passed to the output_metric function.
  
      euclidean_output: bool
          Whether to use the faster code specialised for euclidean output metrics
  
      parallel: bool (optional, default False)
          Whether to run the computation using numba parallel.
          Running in parallel is non-deterministic, and is not used
          if a random seed has been set, to ensure reproducibility.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized of ``graph`` into an ``n_components`` dimensional
          euclidean space.
  
      aux_data: dict
          Auxiliary output returned with the embedding. When densMAP extension
          is turned on, this dictionary includes local radii in the original
          data (``rad_orig``) and in the embedding (``rad_emb``).
      """
      graph = graph.tocoo()
      graph.sum_duplicates()
      n_vertices = graph.shape[1]
  
      # For smaller datasets we can use more epochs
      if graph.shape[0] <= 10000:
          default_epochs = 500
      else:
          default_epochs = 200
  
      # Use more epochs for densMAP
      if densmap:
          default_epochs += 200
  
      # Get the maximum epoch to reach
      n_epochs_max = max(n_epochs) if isinstance(n_epochs, list) else n_epochs
  
      if n_epochs_max is None:
          n_epochs_max = default_epochs
  
      if n_epochs_max > 10:
          graph.data[graph.data < (graph.data.max() / float(n_epochs_max))] = 0.0
      else:
          graph.data[graph.data < (graph.data.max() / float(default_epochs))] = 0.0
  
      graph.eliminate_zeros()
  
      if isinstance(init, str) and init == "random":
          embedding = random_state.uniform(
              low=-10.0, high=10.0, size=(graph.shape[0], n_components)
          ).astype(np.float32)
      elif isinstance(init, str) and init == "spectral":
          # We add a little noise to avoid local minima for optimization to come
          initialisation = spectral_layout(
              data,
              graph,
              n_components,
              random_state,
              metric=metric,
              metric_kwds=metric_kwds,
          )
          expansion = 10.0 / np.abs(initialisation).max()
          embedding = (initialisation * expansion).astype(
              np.float32
          ) + random_state.normal(
              scale=0.0001, size=[graph.shape[0], n_components]
          ).astype(
              np.float32
          )
      else:
          init_data = np.array(init)
          if len(init_data.shape) == 2:
              if np.unique(init_data, axis=0).shape[0] < init_data.shape[0]:
                  tree = KDTree(init_data)
                  dist, ind = tree.query(init_data, k=2)
                  nndist = np.mean(dist[:, 1])
                  embedding = init_data + random_state.normal(
                      scale=0.001 * nndist, size=init_data.shape
                  ).astype(np.float32)
              else:
                  embedding = init_data
  
      epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs_max)
  
      head = graph.row
      tail = graph.col
      weight = graph.data
  
      rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
      aux_data = {}
  
      if densmap or output_dens:
          if verbose:
              print(ts() + " Computing original densities")
  
          dists = densmap_kwds["graph_dists"]
  
          mu_sum = np.zeros(n_vertices, dtype=np.float32)
          ro = np.zeros(n_vertices, dtype=np.float32)
          for i in range(len(head)):
              j = head[i]
              k = tail[i]
  
              D = dists[j, k] * dists[j, k]  # match sq-Euclidean used for embedding
              mu = graph.data[i]
  
              ro[j] += mu * D
              ro[k] += mu * D
              mu_sum[j] += mu
              mu_sum[k] += mu
  
          epsilon = 1e-8
          ro = np.log(epsilon + (ro / mu_sum))
  
          if densmap:
              R = (ro - np.mean(ro)) / np.std(ro)
              densmap_kwds["mu"] = graph.data
              densmap_kwds["mu_sum"] = mu_sum
              densmap_kwds["R"] = R
  
          if output_dens:
              aux_data["rad_orig"] = ro
  
      embedding = (
          10.0
          * (embedding - np.min(embedding, 0))
          / (np.max(embedding, 0) - np.min(embedding, 0))
      ).astype(np.float32, order="C")
  
      if euclidean_output:
          embedding = optimize_layout_euclidean(
              embedding,
              embedding,
              head,
              tail,
              n_epochs,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              initial_alpha,
              negative_sample_rate,
              parallel=parallel,
              verbose=verbose,
              densmap=densmap,
              densmap_kwds=densmap_kwds,
              move_other=True,
          )
      else:
          embedding = optimize_layout_generic(
              embedding,
              embedding,
              head,
              tail,
              n_epochs,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              initial_alpha,
              negative_sample_rate,
              output_metric,
              tuple(output_metric_kwds.values()),
              verbose=verbose,
              move_other=True,
          )
  
      if isinstance(embedding, list):
          aux_data["embedding_list"] = embedding
          embedding = embedding[-1].copy()
  
      if output_dens:
          if verbose:
              print(ts() + " Computing embedding densities")
  
          # Compute graph in embedding
          (knn_indices, knn_dists, rp_forest,) = nearest_neighbors(
              embedding,
              densmap_kwds["n_neighbors"],
              "euclidean",
              {},
              False,
              random_state,
              verbose=verbose,
          )
  
          emb_graph, emb_sigmas, emb_rhos, emb_dists = fuzzy_simplicial_set(
              embedding,
              densmap_kwds["n_neighbors"],
              random_state,
              "euclidean",
              {},
              knn_indices,
              knn_dists,
              verbose=verbose,
              return_dists=True,
          )
  
          emb_graph = emb_graph.tocoo()
          emb_graph.sum_duplicates()
          emb_graph.eliminate_zeros()
  
          n_vertices = emb_graph.shape[1]
  
          mu_sum = np.zeros(n_vertices, dtype=np.float32)
          re = np.zeros(n_vertices, dtype=np.float32)
  
          head = emb_graph.row
          tail = emb_graph.col
          for i in range(len(head)):
              j = head[i]
              k = tail[i]
  
              D = emb_dists[j, k]
              mu = emb_graph.data[i]
  
              re[j] += mu * D
              re[k] += mu * D
              mu_sum[j] += mu
              mu_sum[k] += mu
  
          epsilon = 1e-8
          re = np.log(epsilon + (re / mu_sum))
  
          aux_data["rad_emb"] = re
  
      return embedding, aux_data
  
  
  @numba.njit()
  def init_transform(indices, weights, embedding):
      """Given indices and weights and an original embeddings
      initialize the positions of new points relative to the
      indices and weights (of their neighbors in the source data).
  
      Parameters
      ----------
      indices: array of shape (n_new_samples, n_neighbors)
          The indices of the neighbors of each new sample
  
      weights: array of shape (n_new_samples, n_neighbors)
          The membership strengths of associated 1-simplices
          for each of the new samples.
  
      embedding: array of shape (n_samples, dim)
          The original embedding of the source data.
  
      Returns
      -------
      new_embedding: array of shape (n_new_samples, dim)
          An initial embedding of the new sample points.
      """
      result = np.zeros((indices.shape[0], embedding.shape[1]), dtype=np.float32)
  
      for i in range(indices.shape[0]):
          for j in range(indices.shape[1]):
              for d in range(embedding.shape[1]):
                  result[i, d] += weights[i, j] * embedding[indices[i, j], d]
  
      return result
  
  
  def init_graph_transform(graph, embedding):
      """Given a bipartite graph representing the 1-simplices and strengths between the
       new points and the original data set along with an embedding of the original points
      initialize the positions of new points relative to the strengths (of their neighbors in the source data).
  
      If a point is in our original data set it embeds at the original points coordinates.
      If a point has no neighbours in our original dataset it embeds as the np.nan vector.
      Otherwise a point is the weighted average of it's neighbours embedding locations.
  
      Parameters
      ----------
      graph: csr_matrix (n_new_samples, n_samples)
          A matrix indicating the the 1-simplices and their associated strengths.  These strengths should
          be values between zero and one and not normalized.  One indicating that the new point was identical
          to one of our original points.
  
      embedding: array of shape (n_samples, dim)
          The original embedding of the source data.
  
      Returns
      -------
      new_embedding: array of shape (n_new_samples, dim)
          An initial embedding of the new sample points.
      """
      result = np.zeros((graph.shape[0], embedding.shape[1]), dtype=np.float32)
  
      for row_index in range(graph.shape[0]):
          num_neighbours = len(graph[row_index].indices)
          if num_neighbours == 0:
              result[row_index] = np.nan
              continue
          row_sum = np.sum(graph[row_index])
          for col_index in graph[row_index].indices:
              if graph[row_index, col_index] == 1:
                  result[row_index, :] = embedding[col_index, :]
                  break
              for d in range(embedding.shape[1]):
                  result[row_index, d] += (
                      graph[row_index, col_index] / row_sum * embedding[col_index, d]
                  )
  
      return result
  
  
  @numba.njit()
  def init_update(current_init, n_original_samples, indices):
      for i in range(n_original_samples, indices.shape[0]):
          n = 0
          for j in range(indices.shape[1]):
              for d in range(current_init.shape[1]):
                  if indices[i, j] < n_original_samples:
                      n += 1
                      current_init[i, d] += current_init[indices[i, j], d]
          for d in range(current_init.shape[1]):
              current_init[i, d] /= n
  
      return
  
  
  def find_ab_params(spread, min_dist):
      """Fit a, b params for the differentiable curve used in lower
      dimensional fuzzy simplicial complex construction. We want the
      smooth curve (from a pre-defined family with simple gradient) that
      best matches an offset exponential decay.
      """
  
      def curve(x, a, b):
          return 1.0 / (1.0 + a * x ** (2 * b))
  
      xv = np.linspace(0, spread * 3, 300)
      yv = np.zeros(xv.shape)
      yv[xv < min_dist] = 1.0
      yv[xv >= min_dist] = np.exp(-(xv[xv >= min_dist] - min_dist) / spread)
      params, covar = curve_fit(curve, xv, yv)
      return params[0], params[1]
  
  
  class UMAP(BaseEstimator):
      """Uniform Manifold Approximation and Projection
  
      Finds a low dimensional embedding of the data that approximates
      an underlying manifold.
  
      Parameters
      ----------
      n_neighbors: float (optional, default 15)
          The size of local neighborhood (in terms of number of neighboring
          sample points) used for manifold approximation. Larger values
          result in more global views of the manifold, while smaller
          values result in more local data being preserved. In general
          values should be in the range 2 to 100.
  
      n_components: int (optional, default 2)
          The dimension of the space to embed into. This defaults to 2 to
          provide easy visualization, but can reasonably be set to any
          integer value in the range 2 to 100.
  
      metric: string or function (optional, default 'euclidean')
          The metric to use to compute distances in high dimensional space.
          If a string is passed it must match a valid predefined metric. If
          a general metric is required a function that takes two 1d arrays and
          returns a float can be provided. For performance purposes it is
          required that this be a numba jit'd function. Valid string metrics
          include:
              * euclidean
              * manhattan
              * chebyshev
              * minkowski
              * canberra
              * braycurtis
              * mahalanobis
              * wminkowski
              * seuclidean
              * cosine
              * correlation
              * haversine
              * hamming
              * jaccard
              * dice
              * russelrao
              * kulsinski
              * ll_dirichlet
              * hellinger
              * rogerstanimoto
              * sokalmichener
              * sokalsneath
              * yule
          Metrics that take arguments (such as minkowski, mahalanobis etc.)
          can have arguments passed via the metric_kwds dictionary. At this
          time care must be taken and dictionary elements must be ordered
          appropriately; this will hopefully be fixed in the future.
  
      n_epochs: int (optional, default None)
          The number of training epochs to be used in optimizing the
          low dimensional embedding. Larger values result in more accurate
          embeddings. If None is specified a value will be selected based on
          the size of the input dataset (200 for large datasets, 500 for small).
  
      learning_rate: float (optional, default 1.0)
          The initial learning rate for the embedding optimization.
  
      init: string (optional, default 'spectral')
          How to initialize the low dimensional embedding. Options are:
              * 'spectral': use a spectral embedding of the fuzzy 1-skeleton
              * 'random': assign initial embedding positions at random.
              * A numpy array of initial embedding positions.
  
      min_dist: float (optional, default 0.1)
          The effective minimum distance between embedded points. Smaller values
          will result in a more clustered/clumped embedding where nearby points
          on the manifold are drawn closer together, while larger values will
          result on a more even dispersal of points. The value should be set
          relative to the ``spread`` value, which determines the scale at which
          embedded points will be spread out.
  
      spread: float (optional, default 1.0)
          The effective scale of embedded points. In combination with ``min_dist``
          this determines how clustered/clumped the embedded points are.
  
      low_memory: bool (optional, default True)
          For some datasets the nearest neighbor computation can consume a lot of
          memory. If you find that UMAP is failing due to memory constraints
          consider setting this option to True. This approach is more
          computationally expensive, but avoids excessive memory use.
  
      set_op_mix_ratio: float (optional, default 1.0)
          Interpolate between (fuzzy) union and intersection as the set operation
          used to combine local fuzzy simplicial sets to obtain a global fuzzy
          simplicial sets. Both fuzzy set operations use the product t-norm.
          The value of this parameter should be between 0.0 and 1.0; a value of
          1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy
          intersection.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      repulsion_strength: float (optional, default 1.0)
          Weighting applied to negative samples in low dimensional embedding
          optimization. Values higher than one will result in greater weight
          being given to negative samples.
  
      negative_sample_rate: int (optional, default 5)
          The number of negative samples to select per positive sample
          in the optimization process. Increasing this value will result
          in greater repulsive force being applied, greater optimization
          cost, but slightly more accuracy.
  
      transform_queue_size: float (optional, default 4.0)
          For transform operations (embedding new points using a trained model_
          this will control how aggressively to search for nearest neighbors.
          Larger values will result in slower performance but more accurate
          nearest neighbor evaluation.
  
      a: float (optional, default None)
          More specific parameters controlling the embedding. If None these
          values are set automatically as determined by ``min_dist`` and
          ``spread``.
      b: float (optional, default None)
          More specific parameters controlling the embedding. If None these
          values are set automatically as determined by ``min_dist`` and
          ``spread``.
  
      random_state: int, RandomState instance or None, optional (default: None)
          If int, random_state is the seed used by the random number generator;
          If RandomState instance, random_state is the random number generator;
          If None, the random number generator is the RandomState instance used
          by `np.random`.
  
      metric_kwds: dict (optional, default None)
          Arguments to pass on to the metric, such as the ``p`` value for
          Minkowski distance. If None then no arguments are passed on.
  
      angular_rp_forest: bool (optional, default False)
          Whether to use an angular random projection forest to initialise
          the approximate nearest neighbor search. This can be faster, but is
          mostly on useful for metric that use an angular style distance such
          as cosine, correlation etc. In the case of those metrics angular forests
          will be chosen automatically.
  
      target_n_neighbors: int (optional, default -1)
          The number of nearest neighbors to use to construct the target simplcial
          set. If set to -1 use the ``n_neighbors`` value.
  
      target_metric: string or callable (optional, default 'categorical')
          The metric used to measure distance for a target array is using supervised
          dimension reduction. By default this is 'categorical' which will measure
          distance in terms of whether categories match or are different. Furthermore,
          if semi-supervised is required target values of -1 will be trated as
          unlabelled under the 'categorical' metric. If the target array takes
          continuous values (e.g. for a regression problem) then metric of 'l1'
          or 'l2' is probably more appropriate.
  
      target_metric_kwds: dict (optional, default None)
          Keyword argument to pass to the target metric when performing
          supervised dimension reduction. If None then no arguments are passed on.
  
      target_weight: float (optional, default 0.5)
          weighting factor between data topology and target topology. A value of
          0.0 weights entirely on data, a value of 1.0 weights entirely on target.
          The default of 0.5 balances the weighting equally between data and target.
  
      transform_seed: int (optional, default 42)
          Random seed used for the stochastic aspects of the transform operation.
          This ensures consistency in transform operations.
  
      verbose: bool (optional, default False)
          Controls verbosity of logging.
  
      unique: bool (optional, default False)
          Controls if the rows of your data should be uniqued before being
          embedded.  If you have more duplicates than you have n_neighbour
          you can have the identical data points lying in different regions of
          your space.  It also violates the definition of a metric.
          For to map from internal structures back to your data use the variable
          _unique_inverse_.
  
      densmap: bool (optional, default False)
          Specifies whether the density-augmented objective of densMAP
          should be used for optimization. Turning on this option generates
          an embedding where the local densities are encouraged to be correlated
          with those in the original space. Parameters below with the prefix 'dens'
          further control the behavior of this extension.
  
      dens_lambda: float (optional, default 2.0)
          Controls the regularization weight of the density correlation term
          in densMAP. Higher values prioritize density preservation over the
          UMAP objective, and vice versa for values closer to zero. Setting this
          parameter to zero is equivalent to running the original UMAP algorithm.
  
      dens_frac: float (optional, default 0.3)
          Controls the fraction of epochs (between 0 and 1) where the
          density-augmented objective is used in densMAP. The first
          (1 - dens_frac) fraction of epochs optimize the original UMAP objective
          before introducing the density correlation term.
  
      dens_var_shift: float (optional, default 0.1)
          A small constant added to the variance of local radii in the
          embedding when calculating the density correlation objective to
          prevent numerical instability from dividing by a small number
  
      output_dens: float (optional, default False)
          Determines whether the local radii of the final embedding (an inverse
          measure of local density) are computed and returned in addition to
          the embedding. If set to True, local radii of the original data
          are also included in the output for comparison; the output is a tuple
          (embedding, original local radii, embedding local radii). This option
          can also be used when densmap=False to calculate the densities for
          UMAP embeddings.
  
      disconnection_distance: float (optional, default np.inf or maximal value for bounded distances)
          Disconnect any vertices of distance greater than or equal to disconnection_distance when approximating the
          manifold via our k-nn graph. This is particularly useful in the case that you have a bounded metric.  The
          UMAP assumption that we have a connected manifold can be problematic when you have points that are maximally
          different from all the rest of your data.  The connected manifold assumption will make such points have perfect
          similarity to a random set of other points.  Too many such points will artificially connect your space.
      """
  
      def __init__(
          self,
          n_neighbors=15,
          n_components=2,
          metric="euclidean",
          metric_kwds=None,
          output_metric="euclidean",
          output_metric_kwds=None,
          n_epochs=None,
          learning_rate=1.0,
          init="spectral",
          min_dist=0.1,
          spread=1.0,
          low_memory=True,
          n_jobs=-1,
          set_op_mix_ratio=1.0,
          local_connectivity=1.0,
          repulsion_strength=1.0,
          negative_sample_rate=5,
          transform_queue_size=4.0,
          a=None,
          b=None,
          random_state=None,
          angular_rp_forest=False,
          target_n_neighbors=-1,
          target_metric="categorical",
          target_metric_kwds=None,
          target_weight=0.5,
          transform_seed=42,
          transform_mode="embedding",
          force_approximation_algorithm=False,
          verbose=False,
          unique=False,
          densmap=False,
          dens_lambda=2.0,
          dens_frac=0.3,
          dens_var_shift=0.1,
          output_dens=False,
          disconnection_distance=None,
      ):
          self.n_neighbors = n_neighbors
          self.metric = metric
          self.output_metric = output_metric
          self.target_metric = target_metric
          self.metric_kwds = metric_kwds
          self.output_metric_kwds = output_metric_kwds
          self.n_epochs = n_epochs
          self.init = init
          self.n_components = n_components
          self.repulsion_strength = repulsion_strength
          self.learning_rate = learning_rate
  
          self.spread = spread
          self.min_dist = min_dist
          self.low_memory = low_memory
          self.set_op_mix_ratio = set_op_mix_ratio
          self.local_connectivity = local_connectivity
          self.negative_sample_rate = negative_sample_rate
          self.random_state = random_state
          self.angular_rp_forest = angular_rp_forest
          self.transform_queue_size = transform_queue_size
          self.target_n_neighbors = target_n_neighbors
          self.target_metric = target_metric
          self.target_metric_kwds = target_metric_kwds
          self.target_weight = target_weight
          self.transform_seed = transform_seed
          self.transform_mode = transform_mode
          self.force_approximation_algorithm = force_approximation_algorithm
          self.verbose = verbose
          self.unique = unique
  
          self.densmap = densmap
          self.dens_lambda = dens_lambda
          self.dens_frac = dens_frac
          self.dens_var_shift = dens_var_shift
          self.output_dens = output_dens
          self.disconnection_distance = disconnection_distance
  
          self.n_jobs = n_jobs
  
          self.a = a
          self.b = b
  
      def _validate_parameters(self):
          if self.set_op_mix_ratio < 0.0 or self.set_op_mix_ratio > 1.0:
              raise ValueError("set_op_mix_ratio must be between 0.0 and 1.0")
          if self.repulsion_strength < 0.0:
              raise ValueError("repulsion_strength cannot be negative")
          if self.min_dist > self.spread:
              raise ValueError("min_dist must be less than or equal to spread")
          if self.min_dist < 0.0:
              raise ValueError("min_dist cannot be negative")
          if not isinstance(self.init, str) and not isinstance(self.init, np.ndarray):
              raise ValueError("init must be a string or ndarray")
          if isinstance(self.init, str) and self.init not in (
              "spectral",
              "random",
          ):
              raise ValueError('string init values must be "spectral" or "random"')
          if (
              isinstance(self.init, np.ndarray)
              and self.init.shape[1] != self.n_components
          ):
              raise ValueError("init ndarray must match n_components value")
          if not isinstance(self.metric, str) and not callable(self.metric):
              raise ValueError("metric must be string or callable")
          if self.negative_sample_rate < 0:
              raise ValueError("negative sample rate must be positive")
          if self._initial_alpha < 0.0:
              raise ValueError("learning_rate must be positive")
          if self.n_neighbors < 2:
              raise ValueError("n_neighbors must be greater than 1")
          if self.target_n_neighbors < 2 and self.target_n_neighbors != -1:
              raise ValueError("target_n_neighbors must be greater than 1")
          if not isinstance(self.n_components, int):
              if isinstance(self.n_components, str):
                  raise ValueError("n_components must be an int")
              if self.n_components % 1 != 0:
                  raise ValueError("n_components must be a whole number")
              try:
                  # this will convert other types of int (eg. numpy int64)
                  # to Python int
                  self.n_components = int(self.n_components)
              except ValueError:
                  raise ValueError("n_components must be an int")
          if self.n_components < 1:
              raise ValueError("n_components must be greater than 0")
          self.n_epochs_list = None
          if isinstance(self.n_epochs, list):
              if not all(isinstance(n, int) and n >= 0 for n in self.n_epochs):
                  raise ValueError("n_epochs must be a nonnegative integer or a list of nonnegative integers")
              self.n_epochs_list = self.n_epochs
              self.n_epochs = max(self.n_epochs_list)
          if self.n_epochs is not None and (
                  self.n_epochs < 0 or not isinstance(self.n_epochs, int)
              ):
                  raise ValueError("n_epochs must be a nonnegative integer or a list of nonnegative integers")
          if self.metric_kwds is None:
              self._metric_kwds = {}
          else:
              self._metric_kwds = self.metric_kwds
          if self.output_metric_kwds is None:
              self._output_metric_kwds = {}
          else:
              self._output_metric_kwds = self.output_metric_kwds
          if self.target_metric_kwds is None:
              self._target_metric_kwds = {}
          else:
              self._target_metric_kwds = self.target_metric_kwds
          # check sparsity of data upfront to set proper _input_distance_func &
          # save repeated checks later on
          if scipy.sparse.isspmatrix_csr(self._raw_data):
              self._sparse_data = True
          else:
              self._sparse_data = False
          # set input distance metric & inverse_transform distance metric
          if callable(self.metric):
              in_returns_grad = self._check_custom_metric(
                  self.metric, self._metric_kwds, self._raw_data
              )
              if in_returns_grad:
                  _m = self.metric
  
                  @numba.njit(fastmath=True)
                  def _dist_only(x, y, *kwds):
                      return _m(x, y, *kwds)[0]
  
                  self._input_distance_func = _dist_only
                  self._inverse_distance_func = self.metric
              else:
                  self._input_distance_func = self.metric
                  self._inverse_distance_func = None
                  warn(
                      "custom distance metric does not return gradient; inverse_transform will be unavailable. "
                      "To enable using inverse_transform method, define a distance function that returns a tuple "
                      "of (distance [float], gradient [np.array])"
                  )
          elif self.metric == "precomputed":
              if self.unique:
                  raise ValueError("unique is poorly defined on a precomputed metric")
              warn("using precomputed metric; inverse_transform will be unavailable")
              self._input_distance_func = self.metric
              self._inverse_distance_func = None
          elif self.metric == "hellinger" and self._raw_data.min() < 0:
              raise ValueError("Metric 'hellinger' does not support negative values")
          elif self.metric in dist.named_distances:
              if self._sparse_data:
                  if self.metric in sparse.sparse_named_distances:
                      self._input_distance_func = sparse.sparse_named_distances[
                          self.metric
                      ]
                  else:
                      raise ValueError(
                          "Metric {} is not supported for sparse data".format(self.metric)
                      )
              else:
                  self._input_distance_func = dist.named_distances[self.metric]
              try:
                  self._inverse_distance_func = dist.named_distances_with_gradients[
                      self.metric
                  ]
              except KeyError:
                  warn(
                      "gradient function is not yet implemented for {} distance metric; "
                      "inverse_transform will be unavailable".format(self.metric)
                  )
                  self._inverse_distance_func = None
          elif self.metric in pynn_named_distances:
              if self._sparse_data:
                  if self.metric in pynn_sparse_named_distances:
                      self._input_distance_func = pynn_sparse_named_distances[self.metric]
                  else:
                      raise ValueError(
                          "Metric {} is not supported for sparse data".format(self.metric)
                      )
              else:
                  self._input_distance_func = pynn_named_distances[self.metric]
  
              warn(
                  "gradient function is not yet implemented for {} distance metric; "
                  "inverse_transform will be unavailable".format(self.metric)
              )
              self._inverse_distance_func = None
          else:
              raise ValueError("metric is neither callable nor a recognised string")
          # set output distance metric
          if callable(self.output_metric):
              out_returns_grad = self._check_custom_metric(
                  self.output_metric, self._output_metric_kwds
              )
              if out_returns_grad:
                  self._output_distance_func = self.output_metric
              else:
                  raise ValueError(
                      "custom output_metric must return a tuple of (distance [float], gradient [np.array])"
                  )
          elif self.output_metric == "precomputed":
              raise ValueError("output_metric cannnot be 'precomputed'")
          elif self.output_metric in dist.named_distances_with_gradients:
              self._output_distance_func = dist.named_distances_with_gradients[
                  self.output_metric
              ]
          elif self.output_metric in dist.named_distances:
              raise ValueError(
                  "gradient function is not yet implemented for {}.".format(
                      self.output_metric
                  )
              )
          else:
              raise ValueError(
                  "output_metric is neither callable nor a recognised string"
              )
          # set angularity for NN search based on metric
          if self.metric in (
              "cosine",
              "correlation",
              "dice",
              "jaccard",
              "ll_dirichlet",
              "hellinger",
          ):
              self.angular_rp_forest = True
  
          if self.n_jobs < -1 or self.n_jobs == 0:
              raise ValueError("n_jobs must be a postive integer, or -1 (for all cores)")
  
          if self.dens_lambda < 0.0:
              raise ValueError("dens_lambda cannot be negative")
          if self.dens_frac < 0.0 or self.dens_frac > 1.0:
              raise ValueError("dens_frac must be between 0.0 and 1.0")
          if self.dens_var_shift < 0.0:
              raise ValueError("dens_var_shift cannot be negative")
  
          self._densmap_kwds = {
              "lambda": self.dens_lambda if self.densmap else 0.0,
              "frac": self.dens_frac if self.densmap else 0.0,
              "var_shift": self.dens_var_shift,
              "n_neighbors": self.n_neighbors,
          }
  
          if self.densmap:
              if self.output_metric not in ("euclidean", "l2"):
                  raise ValueError(
                      "Non-Euclidean output metric not supported for densMAP."
                  )
  
          # This will be used to prune all edges of greater than a fixed value from our knn graph.
          # We have preset defaults described in DISCONNECTION_DISTANCES for our bounded measures.
          # Otherwise a user can pass in their own value.
          if self.disconnection_distance is None:
              self._disconnection_distance = DISCONNECTION_DISTANCES.get(
                  self.metric, np.inf
              )
          elif isinstance(self.disconnection_distance, int) or isinstance(
              self.disconnection_distance, float
          ):
              self._disconnection_distance = self.disconnection_distance
          else:
              raise ValueError("disconnection_distance must either be None or a numeric.")
  
      def _check_custom_metric(self, metric, kwds, data=None):
          # quickly check to determine whether user-defined
          # self.metric/self.output_metric returns both distance and gradient
          if data is not None:
              # if checking the high-dimensional distance metric, test directly on
              # input data so we don't risk violating any assumptions potentially
              # hard-coded in the metric (e.g., bounded; non-negative)
              x, y = data[np.random.randint(0, data.shape[0], 2)]
          else:
              # if checking the manifold distance metric, simulate some data on a
              # reasonable interval with output dimensionality
              x, y = np.random.uniform(low=-10, high=10, size=(2, self.n_components))
  
          if scipy.sparse.issparse(data):
              metric_out = metric(x.indices, x.data, y.indices, y.data, **kwds)
          else:
              metric_out = metric(x, y, **kwds)
          # True if metric returns iterable of length 2, False otherwise
          return hasattr(metric_out, "__iter__") and len(metric_out) == 2
  
      def _populate_combined_params(self, *models):
          self.n_neighbors = flattened([m.n_neighbors for m in models])
          self.metric = flattened([m.metric for m in models])
          self.metric_kwds = flattened([m.metric_kwds for m in models])
          self.output_metric = flattened([m.output_metric for m in models])
  
          self.n_epochs = flattened(
              [m.n_epochs if m.n_epochs is not None else -1 for m in models]
          )
          if all([x == -1 for x in self.n_epochs]):
              self.n_epochs = None
  
          self.init = flattened([m.init for m in models])
          self.n_components = flattened([m.n_components for m in models])
          self.repulsion_strength = flattened([m.repulsion_strength for m in models])
          self.learning_rate = flattened([m.learning_rate for m in models])
  
          self.spread = flattened([m.spread for m in models])
          self.min_dist = flattened([m.min_dist for m in models])
          self.low_memory = flattened([m.low_memory for m in models])
          self.set_op_mix_ratio = flattened([m.set_op_mix_ratio for m in models])
          self.local_connectivity = flattened([m.local_connectivity for m in models])
          self.negative_sample_rate = flattened([m.negative_sample_rate for m in models])
          self.random_state = flattened([m.random_state for m in models])
          self.angular_rp_forest = flattened([m.angular_rp_forest for m in models])
          self.transform_queue_size = flattened([m.transform_queue_size for m in models])
          self.target_n_neighbors = flattened([m.target_n_neighbors for m in models])
          self.target_metric = flattened([m.target_metric for m in models])
          self.target_metric_kwds = flattened([m.target_metric_kwds for m in models])
          self.target_weight = flattened([m.target_weight for m in models])
          self.transform_seed = flattened([m.transform_seed for m in models])
          self.force_approximation_algorithm = flattened(
              [m.force_approximation_algorithm for m in models]
          )
          self.verbose = flattened([m.verbose for m in models])
          self.unique = flattened([m.unique for m in models])
  
          self.densmap = flattened([m.densmap for m in models])
          self.dens_lambda = flattened([m.dens_lambda for m in models])
          self.dens_frac = flattened([m.dens_frac for m in models])
          self.dens_var_shift = flattened([m.dens_var_shift for m in models])
          self.output_dens = flattened([m.output_dens for m in models])
  
          self.a = flattened([m.a for m in models])
          self.b = flattened([m.b for m in models])
  
          self._a = flattened([m._a for m in models])
          self._b = flattened([m._b for m in models])
  
      def __mul__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_intersection(
              self.graph_, other.graph_, 0.5
          )
          result.graph_ = reset_local_connectivity(result.graph_, True)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but multi-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def __add__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_union(self.graph_, other.graph_)
          result.graph_ = reset_local_connectivity(result.graph_, True)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but mult-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def __sub__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_intersection(
              self.graph_, other.graph_, weight=0.5, right_complement=True
          )
          result.graph_ = reset_local_connectivity(result.graph_, False)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but mult-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def fit(self, X, y=None):
          """Fit X into an embedded space.
  
          Optionally use y for supervised dimension reduction.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features) or (n_samples, n_samples)
              If the metric is 'precomputed' X must be a square distance
              matrix. Otherwise it contains a sample per row. If the method
              is 'exact', X may be a sparse matrix of type 'csr', 'csc'
              or 'coo'.
  
          y : array, shape (n_samples)
              A target array for supervised dimension reduction. How this is
              handled is determined by parameters UMAP was instantiated with.
              The relevant attributes are ``target_metric`` and
              ``target_metric_kwds``.
          """
  
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          self._raw_data = X
  
          # Handle all the optional arguments, setting default
          if self.a is None or self.b is None:
              self._a, self._b = find_ab_params(self.spread, self.min_dist)
          else:
              self._a = self.a
              self._b = self.b
  
          if isinstance(self.init, np.ndarray):
              init = check_array(self.init, dtype=np.float32, accept_sparse=False)
          else:
              init = self.init
  
          self._initial_alpha = self.learning_rate
  
          self._validate_parameters()
  
          if self.verbose:
              print(str(self))
  
          self._original_n_threads = numba.get_num_threads()
          if self.n_jobs > 0 and self.n_jobs is not None:
              numba.set_num_threads(self.n_jobs)
  
          # Check if we should unique the data
          # We've already ensured that we aren't in the precomputed case
          if self.unique:
              # check if the matrix is dense
              if self._sparse_data:
                  # Call a sparse unique function
                  index, inverse, counts = csr_unique(X)
              else:
                  index, inverse, counts = np.unique(
                      X,
                      return_index=True,
                      return_inverse=True,
                      return_counts=True,
                      axis=0,
                  )[1:4]
              if self.verbose:
                  print(
                      "Unique=True -> Number of data points reduced from ",
                      X.shape[0],
                      " to ",
                      X[index].shape[0],
                  )
                  most_common = np.argmax(counts)
                  print(
                      "Most common duplicate is",
                      index[most_common],
                      " with a count of ",
                      counts[most_common],
                  )
              # We'll expose an inverse map when unique=True for users to map from our internal structures to their data
              self._unique_inverse_ = inverse
          # If we aren't asking for unique use the full index.
          # This will save special cases later.
          else:
              index = list(range(X.shape[0]))
              inverse = list(range(X.shape[0]))
  
          # Error check n_neighbors based on data size
          if X[index].shape[0] <= self.n_neighbors:
              if X[index].shape[0] == 1:
                  self.embedding_ = np.zeros(
                      (1, self.n_components)
                  )  # needed to sklearn comparability
                  return self
  
              warn(
                  "n_neighbors is larger than the dataset size; truncating to "
                  "X.shape[0] - 1"
              )
              self._n_neighbors = X[index].shape[0] - 1
              if self.densmap:
                  self._densmap_kwds["n_neighbors"] = self._n_neighbors
          else:
              self._n_neighbors = self.n_neighbors
  
          # Note: unless it causes issues for setting 'index', could move this to
          # initial sparsity check above
          if self._sparse_data and not X.has_sorted_indices:
              X.sort_indices()
  
          random_state = check_random_state(self.random_state)
  
          if self.verbose:
              print("Construct fuzzy simplicial set")
  
          if self.metric == "precomputed" and self._sparse_data:
              # For sparse precomputed distance matrices, we just argsort the rows to find
              # nearest neighbors. To make this easier, we expect matrices that are
              # symmetrical (so we can find neighbors by looking at rows in isolation,
              # rather than also having to consider that sample's column too).
              # print("Computing KNNs for sparse precomputed distances...")
              if sparse_tril(X).getnnz() != sparse_triu(X).getnnz():
                  raise ValueError(
                      "Sparse precomputed distance matrices should be symmetrical!"
                  )
              if not np.all(X.diagonal() == 0):
                  raise ValueError("Non-zero distances from samples to themselves!")
              self._knn_indices = np.zeros((X.shape[0], self.n_neighbors), dtype=np.int)
              self._knn_dists = np.zeros(self._knn_indices.shape, dtype=np.float)
              for row_id in range(X.shape[0]):
                  # Find KNNs row-by-row
                  row_data = X[row_id].data
                  row_indices = X[row_id].indices
                  if len(row_data) < self._n_neighbors:
                      raise ValueError(
                          "Some rows contain fewer than n_neighbors distances!"
                      )
                  row_nn_data_indices = np.argsort(row_data)[: self._n_neighbors]
                  self._knn_indices[row_id] = row_indices[row_nn_data_indices]
                  self._knn_dists[row_id] = row_data[row_nn_data_indices]
  
              # Disconnect any vertices farther apart than _disconnection_distance
              disconnected_index = self._knn_dists >= self._disconnection_distance
              self._knn_indices[disconnected_index] = -1
              self._knn_dists[disconnected_index] = np.inf
              edges_removed = disconnected_index.sum()
  
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  X[index],
                  self.n_neighbors,
                  random_state,
                  "precomputed",
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
          # Handle small cases efficiently by computing all distances
          elif X[index].shape[0] < 4096 and not self.force_approximation_algorithm:
              self._small_data = True
              try:
                  # sklearn pairwise_distances fails for callable metric on sparse data
                  _m = self.metric if self._sparse_data else self._input_distance_func
                  dmat = pairwise_distances(X[index], metric=_m, **self._metric_kwds)
              except (ValueError, TypeError) as e:
                  # metric is numba.jit'd or not supported by sklearn,
                  # fallback to pairwise special
  
                  if self._sparse_data:
                      # Get a fresh metric since we are casting to dense
                      if not callable(self.metric):
                          _m = dist.named_distances[self.metric]
                          dmat = dist.pairwise_special_metric(
                              X[index].toarray(),
                              metric=_m,
                              kwds=self._metric_kwds,
                          )
                      else:
                          dmat = dist.pairwise_special_metric(
                              X[index],
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  else:
                      dmat = dist.pairwise_special_metric(
                          X[index],
                          metric=self._input_distance_func,
                          kwds=self._metric_kwds,
                      )
              # set any values greater than disconnection_distance to be np.inf.
              # This will have no effect when _disconnection_distance is not set since it defaults to np.inf.
              edges_removed = np.sum(dmat >= self._disconnection_distance)
              dmat[dmat >= self._disconnection_distance] = np.inf
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  dmat,
                  self._n_neighbors,
                  random_state,
                  "precomputed",
                  self._metric_kwds,
                  None,
                  None,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
          else:
              # Standard case
              self._small_data = False
              # Standard case
              if self._sparse_data and self.metric in pynn_sparse_named_distances:
                  nn_metric = self.metric
              elif not self._sparse_data and self.metric in pynn_named_distances:
                  nn_metric = self.metric
              else:
                  nn_metric = self._input_distance_func
  
              (
                  self._knn_indices,
                  self._knn_dists,
                  self._knn_search_index,
              ) = nearest_neighbors(
                  X[index],
                  self._n_neighbors,
                  nn_metric,
                  self._metric_kwds,
                  self.angular_rp_forest,
                  random_state,
                  self.low_memory,
                  use_pynndescent=True,
                  n_jobs=self.n_jobs,
                  verbose=self.verbose,
              )
  
              # Disconnect any vertices farther apart than _disconnection_distance
              disconnected_index = self._knn_dists >= self._disconnection_distance
              self._knn_indices[disconnected_index] = -1
              self._knn_dists[disconnected_index] = np.inf
              edges_removed = disconnected_index.sum()
  
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  X[index],
                  self.n_neighbors,
                  random_state,
                  nn_metric,
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
  
          # Currently not checking if any duplicate points have differing labels
          # Might be worth throwing a warning...
          if y is not None:
              len_X = len(X) if not self._sparse_data else X.shape[0]
              if len_X != len(y):
                  raise ValueError(
                      "Length of x = {len_x}, length of y = {len_y}, while it must be equal.".format(
                          len_x=len_X, len_y=len(y)
                      )
                  )
              if self.target_metric == "string":
                  y_ = y[index]
              else:
                  y_ = check_array(y, ensure_2d=False)[index]
              if self.target_metric == "categorical":
                  if self.target_weight < 1.0:
                      far_dist = 2.5 * (1.0 / (1.0 - self.target_weight))
                  else:
                      far_dist = 1.0e12
                  self.graph_ = discrete_metric_simplicial_set_intersection(
                      self.graph_, y_, far_dist=far_dist
                  )
              elif self.target_metric in dist.DISCRETE_METRICS:
                  if self.target_weight < 1.0:
                      scale = 2.5 * (1.0 / (1.0 - self.target_weight))
                  else:
                      scale = 1.0e12
                  # self.graph_ = discrete_metric_simplicial_set_intersection(
                  #     self.graph_,
                  #     y_,
                  #     metric=self.target_metric,
                  #     metric_kws=self.target_metric_kwds,
                  #     metric_scale=scale
                  # )
  
                  metric_kws = dist.get_discrete_params(y_, self.target_metric)
  
                  self.graph_ = discrete_metric_simplicial_set_intersection(
                      self.graph_,
                      y_,
                      metric=self.target_metric,
                      metric_kws=metric_kws,
                      metric_scale=scale,
                  )
              else:
                  if len(y_.shape) == 1:
                      y_ = y_.reshape(-1, 1)
                  if self.target_n_neighbors == -1:
                      target_n_neighbors = self._n_neighbors
                  else:
                      target_n_neighbors = self.target_n_neighbors
  
                  # Handle the small case as precomputed as before
                  if y.shape[0] < 4096:
                      try:
                          ydmat = pairwise_distances(
                              y_, metric=self.target_metric, **self._target_metric_kwds
                          )
                      except (TypeError, ValueError):
                          ydmat = dist.pairwise_special_metric(
                              y_,
                              metric=self.target_metric,
                              kwds=self._target_metric_kwds,
                          )
  
                      (target_graph, target_sigmas, target_rhos,) = fuzzy_simplicial_set(
                          ydmat,
                          target_n_neighbors,
                          random_state,
                          "precomputed",
                          self._target_metric_kwds,
                          None,
                          None,
                          False,
                          1.0,
                          1.0,
                          False,
                      )
                  else:
                      # Standard case
                      (target_graph, target_sigmas, target_rhos,) = fuzzy_simplicial_set(
                          y_,
                          target_n_neighbors,
                          random_state,
                          self.target_metric,
                          self._target_metric_kwds,
                          None,
                          None,
                          False,
                          1.0,
                          1.0,
                          False,
                      )
                  # product = self.graph_.multiply(target_graph)
                  # # self.graph_ = 0.99 * product + 0.01 * (self.graph_ +
                  # #                                        target_graph -
                  # #                                        product)
                  # self.graph_ = product
                  self.graph_ = general_simplicial_set_intersection(
                      self.graph_, target_graph, self.target_weight
                  )
                  self.graph_ = reset_local_connectivity(self.graph_)
              self._supervised = True
          else:
              self._supervised = False
  
          if self.densmap or self.output_dens:
              self._densmap_kwds["graph_dists"] = self.graph_dists_
  
          if self.verbose:
              print(ts(), "Construct embedding")
  
          if self.transform_mode == "embedding":
              epochs = self.n_epochs_list if self.n_epochs_list is not None else self.n_epochs
              self.embedding_, aux_data = self._fit_embed_data(
                  self._raw_data[index],
                  epochs,
                  init,
                  random_state,  # JH why raw data?
              )
  
              if self.n_epochs_list is not None:
                  if not "embedding_list" in aux_data:
                      raise KeyError("No list of embedding were found in 'aux_data'. It is likely the"
                                     "layout optimization function doesn't support the list of int for 'n_epochs'.")
                  else:
                      self.embedding_list = aux_data["embedding_list"]
  
              # Assign any points that are fully disconnected from our manifold(s) to have embedding
              # coordinates of np.nan.  These will be filtered by our plotting functions automatically.
              # They also prevent users from being deceived a distance query to one of these points.
              # Might be worth moving this into simplicial_set_embedding or _fit_embed_data
              disconnected_vertices = np.array(self.graph_.sum(axis=1)).flatten() == 0
              if len(disconnected_vertices) > 0:
                  self.embedding_[disconnected_vertices] = np.full(
                      self.n_components, np.nan
                  )
  
              self.embedding_ = self.embedding_[inverse]
              if self.output_dens:
                  self.rad_orig_ = aux_data["rad_orig"][inverse]
                  self.rad_emb_ = aux_data["rad_emb"][inverse]
  
          if self.verbose:
              print(ts() + " Finished embedding")
  
          numba.set_num_threads(self._original_n_threads)
          self._input_hash = joblib.hash(self._raw_data)
  
          return self
  
      def _fit_embed_data(self, X, n_epochs, init, random_state):
          """A method wrapper for simplicial_set_embedding that can be
          replaced by subclasses.
          """
          return simplicial_set_embedding(
              X,
              self.graph_,
              self.n_components,
              self._initial_alpha,
              self._a,
              self._b,
              self.repulsion_strength,
              self.negative_sample_rate,
              n_epochs,
              init,
              random_state,
              self._input_distance_func,
              self._metric_kwds,
              self.densmap,
              self._densmap_kwds,
              self.output_dens,
              self._output_distance_func,
              self._output_metric_kwds,
              self.output_metric in ("euclidean", "l2"),
              self.random_state is None,
              self.verbose,
          )
  
      def fit_transform(self, X, y=None):
          """Fit X into an embedded space and return that transformed
          output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features) or (n_samples, n_samples)
              If the metric is 'precomputed' X must be a square distance
              matrix. Otherwise it contains a sample per row.
  
          y : array, shape (n_samples)
              A target array for supervised dimension reduction. How this is
              handled is determined by parameters UMAP was instantiated with.
              The relevant attributes are ``target_metric`` and
              ``target_metric_kwds``.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Embedding of the training data in low-dimensional space.
  
          or a tuple (X_new, r_orig, r_emb) if ``output_dens`` flag is set,
          which additionally includes:
  
          r_orig: array, shape (n_samples)
              Local radii of data points in the original data space (log-transformed).
  
          r_emb: array, shape (n_samples)
              Local radii of data points in the embedding (log-transformed).
          """
          self.fit(X, y)
          if self.transform_mode == "embedding":
              if self.output_dens:
                  return self.embedding_, self.rad_orig_, self.rad_emb_
              else:
                  return self.embedding_
          elif self.transform_mode == "graph":
              return self.graph_
          else:
              raise ValueError(
                  "Unrecognized transform mode {}; should be one of 'embedding' or 'graph'".format(
                      self.transform_mode
                  )
              )
  
      def transform(self, X):
          """Transform X into the existing embedded space and return that
          transformed output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features)
              New data to be transformed.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Embedding of the new data in low-dimensional space.
          """
          # If we fit just a single instance then error
          if self._raw_data.shape[0] == 1:
              raise ValueError(
                  "Transform unavailable when model was fit with only a single data sample."
              )
          # If we just have the original input then short circuit things
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          x_hash = joblib.hash(X)
          if x_hash == self._input_hash:
              if self.transform_mode == "embedding":
                  return self.embedding_
              elif self.transform_mode == "graph":
                  return self.graph_
              else:
                  raise ValueError(
                      "Unrecognized transform mode {}; should be one of 'embedding' or 'graph'".format(
                          self.transform_mode
                      )
                  )
          if self.densmap:
              raise NotImplementedError(
                  "Transforming data into an existing embedding not supported for densMAP."
              )
  
          # X = check_array(X, dtype=np.float32, order="C", accept_sparse="csr")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          if self.metric == "precomputed":
              warn(
                  "Transforming new data with precomputed metric. "
                  "We are assuming the input data is a matrix of distances from the new points "
                  "to the points in the training set. If the input matrix is sparse, it should "
                  "contain distances from the new points to their nearest neighbours "
                  "or approximate nearest neighbours in the training set."
              )
              assert X.shape[1] == self._raw_data.shape[0]
              if scipy.sparse.issparse(X):
                  indices = np.full(
                      (X.shape[0], self._n_neighbors), dtype=np.int32, fill_value=-1
                  )
                  dists = np.full_like(indices, dtype=np.float32, fill_value=-1)
                  for i in range(X.shape[0]):
                      data_indices = np.argsort(X[i].data)
                      if len(data_indices) < self._n_neighbors:
                          raise ValueError(
                              f"Need at least n_neighbors ({self.n_neighbors}) distances for each row!"
                          )
                      indices[i] = X[i].indices[data_indices[: self._n_neighbors]]
                      dists[i] = X[i].data[data_indices[: self._n_neighbors]]
              else:
                  indices = np.argsort(X, axis=1)[:, : self._n_neighbors].astype(np.int32)
                  dists = np.take_along_axis(X, indices, axis=1)
              assert np.min(indices) >= 0 and np.min(dists) >= 0.0
          elif self._small_data:
              try:
                  # sklearn pairwise_distances fails for callable metric on sparse data
                  _m = self.metric if self._sparse_data else self._input_distance_func
                  dmat = pairwise_distances(
                      X, self._raw_data, metric=_m, **self._metric_kwds
                  )
              except (TypeError, ValueError):
                  # metric is numba.jit'd or not supported by sklearn,
                  # fallback to pairwise special
                  if self._sparse_data:
                      # Get a fresh metric since we are casting to dense
                      if not callable(self.metric):
                          _m = dist.named_distances[self.metric]
                          dmat = dist.pairwise_special_metric(
                              X.toarray(),
                              self._raw_data.toarray(),
                              metric=_m,
                              kwds=self._metric_kwds,
                          )
                      else:
                          dmat = dist.pairwise_special_metric(
                              X,
                              self._raw_data,
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  else:
                      dmat = dist.pairwise_special_metric(
                          X,
                          self._raw_data,
                          metric=self._input_distance_func,
                          kwds=self._metric_kwds,
                      )
              indices = np.argpartition(dmat, self._n_neighbors)[:, : self._n_neighbors]
              dmat_shortened = submatrix(dmat, indices, self._n_neighbors)
              indices_sorted = np.argsort(dmat_shortened)
              indices = submatrix(indices, indices_sorted, self._n_neighbors)
              dists = submatrix(dmat_shortened, indices_sorted, self._n_neighbors)
          else:
              epsilon = 0.24 if self._knn_search_index._angular_trees else 0.12
              indices, dists = self._knn_search_index.query(
                  X, self.n_neighbors, epsilon=epsilon
              )
  
          dists = dists.astype(np.float32, order="C")
          # Remove any nearest neighbours who's distances are greater than our disconnection_distance
          indices[dists >= self._disconnection_distance] = -1
          adjusted_local_connectivity = max(0.0, self.local_connectivity - 1.0)
          sigmas, rhos = smooth_knn_dist(
              dists,
              float(self._n_neighbors),
              local_connectivity=float(adjusted_local_connectivity),
          )
  
          rows, cols, vals, dists = compute_membership_strengths(
              indices, dists, sigmas, rhos, bipartite=True
          )
  
          graph = scipy.sparse.coo_matrix(
              (vals, (rows, cols)), shape=(X.shape[0], self._raw_data.shape[0])
          )
  
          if self.transform_mode == "graph":
              return graph
  
          # This was a very specially constructed graph with constant degree.
          # That lets us do fancy unpacking by reshaping the csr matrix indices
          # and data. Doing so relies on the constant degree assumption!
          # csr_graph = normalize(graph.tocsr(), norm="l1")
          # inds = csr_graph.indices.reshape(X.shape[0], self._n_neighbors)
          # weights = csr_graph.data.reshape(X.shape[0], self._n_neighbors)
          # embedding = init_transform(inds, weights, self.embedding_)
          # This is less fast code than the above numba.jit'd code.
          # It handles the fact that our nearest neighbour graph can now contain variable numbers of vertices.
          csr_graph = graph.tocsr()
          csr_graph.eliminate_zeros()
          embedding = init_graph_transform(csr_graph, self.embedding_)
  
          if self.n_epochs is None:
              # For smaller datasets we can use more epochs
              if graph.shape[0] <= 10000:
                  n_epochs = 100
              else:
                  n_epochs = 30
          else:
              n_epochs = int(self.n_epochs // 3.0)
  
          graph.data[graph.data < (graph.data.max() / float(n_epochs))] = 0.0
          graph.eliminate_zeros()
  
          epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs)
  
          head = graph.row
          tail = graph.col
          weight = graph.data
  
          # optimize_layout = make_optimize_layout(
          #     self._output_distance_func,
          #     tuple(self.output_metric_kwds.values()),
          # )
  
          if self.output_metric == "euclidean":
              embedding = optimize_layout_euclidean(
                  embedding,
                  self.embedding_.astype(np.float32, copy=True),  # Fixes #179 & #217,
                  head,
                  tail,
                  n_epochs,
                  graph.shape[1],
                  epochs_per_sample,
                  self._a,
                  self._b,
                  rng_state,
                  self.repulsion_strength,
                  self._initial_alpha / 4.0,
                  self.negative_sample_rate,
                  self.random_state is None,
                  verbose=self.verbose,
              )
          else:
              embedding = optimize_layout_generic(
                  embedding,
                  self.embedding_.astype(np.float32, copy=True),  # Fixes #179 & #217
                  head,
                  tail,
                  n_epochs,
                  graph.shape[1],
                  epochs_per_sample,
                  self._a,
                  self._b,
                  rng_state,
                  self.repulsion_strength,
                  self._initial_alpha / 4.0,
                  self.negative_sample_rate,
                  self._output_distance_func,
                  tuple(self._output_metric_kwds.values()),
                  verbose=self.verbose,
              )
  
          return embedding
  
      def inverse_transform(self, X):
          """Transform X in the existing embedded space back into the input
          data space and return that transformed output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_components)
              New points to be inverse transformed.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_features)
              Generated data points new data in data space.
          """
  
          if self._sparse_data:
              raise ValueError("Inverse transform not available for sparse input.")
          elif self._inverse_distance_func is None:
              raise ValueError("Inverse transform not available for given metric.")
          elif self.densmap:
              raise ValueError("Inverse transform not available for densMAP.")
          elif self.n_components >= 8:
              warn(
                  "Inverse transform works best with low dimensional embeddings."
                  " Results may be poor, or this approach to inverse transform"
                  " may fail altogether! If you need a high dimensional latent"
                  " space and inverse transform operations consider using an"
                  " autoencoder."
              )
          elif self.transform_mode == "graph":
              raise ValueError(
                  "Inverse transform not available for transform_mode = 'graph'"
              )
  
          X = check_array(X, dtype=np.float32, order="C")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          # build Delaunay complex (Does this not assume a roughly euclidean output metric)?
          deltri = scipy.spatial.Delaunay(
              self.embedding_, incremental=True, qhull_options="QJ"
          )
          neighbors = deltri.simplices[deltri.find_simplex(X)]
          adjmat = scipy.sparse.lil_matrix(
              (self.embedding_.shape[0], self.embedding_.shape[0]), dtype=int
          )
          for i in np.arange(0, deltri.simplices.shape[0]):
              for j in deltri.simplices[i]:
                  if j < self.embedding_.shape[0]:
                      idx = deltri.simplices[i][
                          deltri.simplices[i] < self.embedding_.shape[0]
                      ]
                      adjmat[j, idx] = 1
                      adjmat[idx, j] = 1
  
          adjmat = scipy.sparse.csr_matrix(adjmat)
  
          min_vertices = min(self._raw_data.shape[-1], self._raw_data.shape[0])
  
          neighborhood = [
              breadth_first_search(adjmat, v[0], min_vertices=min_vertices)
              for v in neighbors
          ]
          if callable(self.output_metric):
              # need to create another numba.jit-able wrapper for callable
              # output_metrics that return a tuple (already checked that it does
              # during param validation in `fit` method)
              _out_m = self.output_metric
  
              @numba.njit(fastmath=True)
              def _output_dist_only(x, y, *kwds):
                  return _out_m(x, y, *kwds)[0]
  
              dist_only_func = _output_dist_only
          elif self.output_metric in dist.named_distances.keys():
              dist_only_func = dist.named_distances[self.output_metric]
          else:
              # shouldn't really ever get here because of checks already performed,
              # but works as a failsafe in case attr was altered manually after fitting
              raise ValueError(
                  "Unrecognized output metric: {}".format(self.output_metric)
              )
  
          dist_args = tuple(self._output_metric_kwds.values())
          distances = [
              np.array(
                  [
                      dist_only_func(X[i], self.embedding_[nb], *dist_args)
                      for nb in neighborhood[i]
                  ]
              )
              for i in range(X.shape[0])
          ]
          idx = np.array([np.argsort(e)[:min_vertices] for e in distances])
  
          dists_output_space = np.array(
              [distances[i][idx[i]] for i in range(len(distances))]
          )
          indices = np.array([neighborhood[i][idx[i]] for i in range(len(neighborhood))])
  
          rows, cols, distances = np.array(
              [
                  [i, indices[i, j], dists_output_space[i, j]]
                  for i in range(indices.shape[0])
                  for j in range(min_vertices)
              ]
          ).T
  
          # calculate membership strength of each edge
          weights = 1 / (1 + self._a * distances ** (2 * self._b))
  
          # compute 1-skeleton
          # convert 1-skeleton into coo_matrix adjacency matrix
          graph = scipy.sparse.coo_matrix(
              (weights, (rows, cols)), shape=(X.shape[0], self._raw_data.shape[0])
          )
  
          # That lets us do fancy unpacking by reshaping the csr matrix indices
          # and data. Doing so relies on the constant degree assumption!
          # csr_graph = graph.tocsr()
          csr_graph = normalize(graph.tocsr(), norm="l1")
          inds = csr_graph.indices.reshape(X.shape[0], min_vertices)
          weights = csr_graph.data.reshape(X.shape[0], min_vertices)
          inv_transformed_points = init_transform(inds, weights, self._raw_data)
  
          if self.n_epochs is None:
              # For smaller datasets we can use more epochs
              if graph.shape[0] <= 10000:
                  n_epochs = 100
              else:
                  n_epochs = 30
          else:
              n_epochs = int(self.n_epochs // 3.0)
  
          # graph.data[graph.data < (graph.data.max() / float(n_epochs))] = 0.0
          # graph.eliminate_zeros()
  
          epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs)
  
          head = graph.row
          tail = graph.col
          weight = graph.data
  
          inv_transformed_points = optimize_layout_inverse(
              inv_transformed_points,
              self._raw_data,
              head,
              tail,
              weight,
              self._sigmas,
              self._rhos,
              n_epochs,
              graph.shape[1],
              epochs_per_sample,
              self._a,
              self._b,
              rng_state,
              self.repulsion_strength,
              self._initial_alpha / 4.0,
              self.negative_sample_rate,
              self._inverse_distance_func,
              tuple(self._metric_kwds.values()),
              verbose=self.verbose,
          )
  
          return inv_transformed_points
  
      def update(self, X):
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          original_size = self._raw_data.shape[0]
  
          if self.metric == "precomputed":
              raise ValueError("Update does not currently support precomputed metrics")
          if self._supervised:
              raise ValueError("Updating supervised models is not currently " "supported")
  
          if self._small_data:
  
              if self._sparse_data:
                  self._raw_data = scipy.sparse.vstack([self._raw_data, X])
              else:
                  self._raw_data = np.vstack([self._raw_data, X])
  
              if self._raw_data.shape[0] < 4096:
                  # still small data
                  try:
                      # sklearn pairwise_distances fails for callable metric on sparse data
                      _m = self.metric if self._sparse_data else self._input_distance_func
                      dmat = pairwise_distances(
                          self._raw_data, metric=_m, **self._metric_kwds
                      )
                  except (ValueError, TypeError) as e:
                      # metric is numba.jit'd or not supported by sklearn,
                      # fallback to pairwise special
  
                      if self._sparse_data:
                          # Get a fresh metric since we are casting to dense
                          if not callable(self.metric):
                              _m = dist.named_distances[self.metric]
                              dmat = dist.pairwise_special_metric(
                                  self._raw_data.toarray(),
                                  metric=_m,
                                  kwds=self._metric_kwds,
                              )
                          else:
                              dmat = dist.pairwise_special_metric(
                                  self._raw_data,
                                  metric=self._input_distance_func,
                                  kwds=self._metric_kwds,
                              )
                      else:
                          dmat = dist.pairwise_special_metric(
                              self._raw_data,
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                      dmat,
                      self._n_neighbors,
                      random_state,
                      "precomputed",
                      self._metric_kwds,
                      None,
                      None,
                      self.angular_rp_forest,
                      self.set_op_mix_ratio,
                      self.local_connectivity,
                      True,
                      self.verbose,
                  )
                  knn_indices = np.argsort(dmat)[:, : self.n_neighbors]
              else:
                  # now large data
                  self._small_data = False
                  if self._sparse_data and self.metric in pynn_sparse_named_distances:
                      nn_metric = self.metric
                  elif not self._sparse_data and self.metric in pynn_named_distances:
                      nn_metric = self.metric
                  else:
                      nn_metric = self._input_distance_func
  
                  (
                      self._knn_indices,
                      self._knn_dists,
                      self._knn_search_index,
                  ) = nearest_neighbors(
                      self._raw_data,
                      self._n_neighbors,
                      nn_metric,
                      self._metric_kwds,
                      self.angular_rp_forest,
                      random_state,
                      self.low_memory,
                      use_pynndescent=True,
                      n_jobs=self.n_jobs,
                      verbose=self.verbose,
                  )
  
                  self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                      self._raw_data,
                      self.n_neighbors,
                      random_state,
                      nn_metric,
                      self._metric_kwds,
                      self._knn_indices,
                      self._knn_dists,
                      self.angular_rp_forest,
                      self.set_op_mix_ratio,
                      self.local_connectivity,
                      True,
                      self.verbose,
                  )
                  knn_indices = self._knn_indices
  
              init = np.zeros(
                  (self._raw_data.shape[0], self.n_components), dtype=np.float32
              )
              init[:original_size] = self.embedding_
  
              init_update(init, original_size, knn_indices)
              if self.n_epochs is None:
                  n_epochs = 0
              else:
                  n_epochs = self.n_epochs
  
              self.embedding_, aux_data = simplicial_set_embedding(
                  self._raw_data,
                  self.graph_,
                  self.n_components,
                  self._initial_alpha,
                  self._a,
                  self._b,
                  self.repulsion_strength,
                  self.negative_sample_rate,
                  n_epochs,
                  init,
                  random_state,
                  self._input_distance_func,
                  self._metric_kwds,
                  self.densmap,
                  self._densmap_kwds,
                  self.output_dens,
                  self._output_distance_func,
                  self._output_metric_kwds,
                  self.output_metric in ("euclidean", "l2"),
                  self.random_state is None,
                  self.verbose,
              )
  
          else:
              self._knn_search_index.update(X)
              self._raw_data = self._knn_search_index._raw_data
              (
                  self._knn_indices,
                  self._knn_dists,
              ) = self._knn_search_index.neighbor_graph
  
              if self._sparse_data and self.metric in pynn_sparse_named_distances:
                  nn_metric = self.metric
              elif not self._sparse_data and self.metric in pynn_named_distances:
                  nn_metric = self.metric
              else:
                  nn_metric = self._input_distance_func
  
              self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                  self._raw_data,
                  self.n_neighbors,
                  random_state,
                  nn_metric,
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
              )
  
              init = np.zeros(
                  (self._raw_data.shape[0], self.n_components), dtype=np.float32
              )
              init[:original_size] = self.embedding_
              init_update(init, original_size, self._knn_indices)
  
              if self.n_epochs is None:
                  n_epochs = 0
              else:
                  n_epochs = self.n_epochs
  
              self.embedding_, aux_data = simplicial_set_embedding(
                  self._raw_data,
                  self.graph_,
                  self.n_components,
                  self._initial_alpha,
                  self._a,
                  self._b,
                  self.repulsion_strength,
                  self.negative_sample_rate,
                  n_epochs,
                  init,
                  random_state,
                  self._input_distance_func,
                  self._metric_kwds,
                  self.densmap,
                  self._densmap_kwds,
                  self.output_dens,
                  self._output_distance_func,
                  self._output_metric_kwds,
                  self.output_metric in ("euclidean", "l2"),
                  self.random_state is None,
                  self.verbose,
              )
  
          if self.output_dens:
              self.rad_orig_ = aux_data["rad_orig"]
              self.rad_emb_ = aux_data["rad_emb"]
  
files: umap/umap_.py
files: # Author: Leland McInnes <leland.mcinnes@gmail.com>
  #
  # License: BSD 3 clause
  from __future__ import print_function
  
  import locale
  from warnings import warn
  import time
  
  from scipy.optimize import curve_fit
  from sklearn.base import BaseEstimator
  from sklearn.utils import check_random_state, check_array
  from sklearn.utils.validation import check_is_fitted
  from sklearn.metrics import pairwise_distances
  from sklearn.preprocessing import normalize
  from sklearn.neighbors import KDTree
  
  try:
      import joblib
  except ImportError:
      # sklearn.externals.joblib is deprecated in 0.21, will be removed in 0.23
      from sklearn.externals import joblib
  
  import numpy as np
  import scipy.sparse
  from scipy.sparse import tril as sparse_tril, triu as sparse_triu
  import scipy.sparse.csgraph
  import numba
  
  import umap.distances as dist
  
  import umap.sparse as sparse
  
  from umap.utils import (
      submatrix,
      ts,
      csr_unique,
      fast_knn_indices,
  )
  from umap.spectral import spectral_layout
  from umap.layouts import (
      optimize_layout_euclidean,
      optimize_layout_generic,
      optimize_layout_inverse,
  )
  
  from pynndescent import NNDescent
  from pynndescent.distances import named_distances as pynn_named_distances
  from pynndescent.sparse import sparse_named_distances as pynn_sparse_named_distances
  
  locale.setlocale(locale.LC_NUMERIC, "C")
  
  INT32_MIN = np.iinfo(np.int32).min + 1
  INT32_MAX = np.iinfo(np.int32).max - 1
  
  SMOOTH_K_TOLERANCE = 1e-5
  MIN_K_DIST_SCALE = 1e-3
  NPY_INFINITY = np.inf
  
  DISCONNECTION_DISTANCES = {
      "correlation": 2,
      "cosine": 2,
      "hellinger": 1,
      "jaccard": 1,
      "dice": 1,
  }
  
  
  def flatten_iter(container):
      for i in container:
          if isinstance(i, (list, tuple)):
              for j in flatten_iter(i):
                  yield j
          else:
              yield i
  
  
  def flattened(container):
      return tuple(flatten_iter(container))
  
  
  def breadth_first_search(adjmat, start, min_vertices):
      explored = []
      queue = [start]
      levels = {}
      levels[start] = 0
      max_level = np.inf
      visited = [start]
  
      while queue:
          node = queue.pop(0)
          explored.append(node)
          if max_level == np.inf and len(explored) > min_vertices:
              max_level = max(levels.values())
  
          if levels[node] + 1 < max_level:
              neighbors = adjmat[node].indices
              for neighbour in neighbors:
                  if neighbour not in visited:
                      queue.append(neighbour)
                      visited.append(neighbour)
  
                      levels[neighbour] = levels[node] + 1
  
      return np.array(explored)
  
  
  def raise_disconnected_warning(
      edges_removed,
      vertices_disconnected,
      disconnection_distance,
      total_rows,
      threshold=0.1,
      verbose=False,
  ):
      """A simple wrapper function to avoid large amounts of code repetition."""
      if verbose & (vertices_disconnected == 0) & (edges_removed > 0):
          print(
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.  "
              f"This is not a problem as no vertices were disconnected."
          )
      elif (vertices_disconnected > 0) & (
          vertices_disconnected <= threshold * total_rows
      ):
          warn(
              f"A few of your vertices were disconnected from the manifold.  This shouldn't cause problems.\n"
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.\n"
              f"It has only fully disconnected {vertices_disconnected} vertices.\n"
              f"Use umap.utils.disconnected_vertices() to identify them.",
          )
      elif vertices_disconnected > threshold * total_rows:
          warn(
              f"A large number of your vertices were disconnected from the manifold.\n"
              f"Disconnection_distance = {disconnection_distance} has removed {edges_removed} edges.\n"
              f"It has fully disconnected {vertices_disconnected} vertices.\n"
              f"You might consider using find_disconnected_points() to find and remove these points from your data.\n"
              f"Use umap.utils.disconnected_vertices() to identify them.",
          )
  
  
  @numba.njit(
      locals={
          "psum": numba.types.float32,
          "lo": numba.types.float32,
          "mid": numba.types.float32,
          "hi": numba.types.float32,
      },
      fastmath=True,
  )  # benchmarking `parallel=True` shows it to *decrease* performance
  def smooth_knn_dist(distances, k, n_iter=64, local_connectivity=1.0, bandwidth=1.0):
      """Compute a continuous version of the distance to the kth nearest
      neighbor. That is, this is similar to knn-distance but allows continuous
      k values rather than requiring an integral k. In essence we are simply
      computing the distance such that the cardinality of fuzzy set we generate
      is k.
  
      Parameters
      ----------
      distances: array of shape (n_samples, n_neighbors)
          Distances to nearest neighbors for each samples. Each row should be a
          sorted list of distances to a given samples nearest neighbors.
  
      k: float
          The number of nearest neighbors to approximate for.
  
      n_iter: int (optional, default 64)
          We need to binary search for the correct distance value. This is the
          max number of iterations to use in such a search.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      bandwidth: float (optional, default 1)
          The target bandwidth of the kernel, larger values will produce
          larger return values.
  
      Returns
      -------
      knn_dist: array of shape (n_samples,)
          The distance to kth nearest neighbor, as suitably approximated.
  
      nn_dist: array of shape (n_samples,)
          The distance to the 1st nearest neighbor for each point.
      """
      target = np.log2(k) * bandwidth
      rho = np.zeros(distances.shape[0], dtype=np.float32)
      result = np.zeros(distances.shape[0], dtype=np.float32)
  
      mean_distances = np.mean(distances)
  
      for i in range(distances.shape[0]):
          lo = 0.0
          hi = NPY_INFINITY
          mid = 1.0
  
          # TODO: This is very inefficient, but will do for now. FIXME
          ith_distances = distances[i]
          non_zero_dists = ith_distances[ith_distances > 0.0]
          if non_zero_dists.shape[0] >= local_connectivity:
              index = int(np.floor(local_connectivity))
              interpolation = local_connectivity - index
              if index > 0:
                  rho[i] = non_zero_dists[index - 1]
                  if interpolation > SMOOTH_K_TOLERANCE:
                      rho[i] += interpolation * (
                          non_zero_dists[index] - non_zero_dists[index - 1]
                      )
              else:
                  rho[i] = interpolation * non_zero_dists[0]
          elif non_zero_dists.shape[0] > 0:
              rho[i] = np.max(non_zero_dists)
  
          for n in range(n_iter):
  
              psum = 0.0
              for j in range(1, distances.shape[1]):
                  d = distances[i, j] - rho[i]
                  if d > 0:
                      psum += np.exp(-(d / mid))
                  else:
                      psum += 1.0
  
              if np.fabs(psum - target) < SMOOTH_K_TOLERANCE:
                  break
  
              if psum > target:
                  hi = mid
                  mid = (lo + hi) / 2.0
              else:
                  lo = mid
                  if hi == NPY_INFINITY:
                      mid *= 2
                  else:
                      mid = (lo + hi) / 2.0
  
          result[i] = mid
  
          # TODO: This is very inefficient, but will do for now. FIXME
          if rho[i] > 0.0:
              mean_ith_distances = np.mean(ith_distances)
              if result[i] < MIN_K_DIST_SCALE * mean_ith_distances:
                  result[i] = MIN_K_DIST_SCALE * mean_ith_distances
          else:
              if result[i] < MIN_K_DIST_SCALE * mean_distances:
                  result[i] = MIN_K_DIST_SCALE * mean_distances
  
      return result, rho
  
  
  def nearest_neighbors(
      X,
      n_neighbors,
      metric,
      metric_kwds,
      angular,
      random_state,
      low_memory=True,
      use_pynndescent=True,
      n_jobs=-1,
      verbose=False,
  ):
      """Compute the ``n_neighbors`` nearest points for each data point in ``X``
      under ``metric``. This may be exact, but more likely is approximated via
      nearest neighbor descent.
  
      Parameters
      ----------
      X: array of shape (n_samples, n_features)
          The input data to compute the k-neighbor graph of.
  
      n_neighbors: int
          The number of nearest neighbors to compute for each sample in ``X``.
  
      metric: string or callable
          The metric to use for the computation.
  
      metric_kwds: dict
          Any arguments to pass to the metric computation function.
  
      angular: bool
          Whether to use angular rp trees in NN approximation.
  
      random_state: np.random state
          The random state to use for approximate NN computations.
  
      low_memory: bool (optional, default True)
          Whether to pursue lower memory NNdescent.
  
      verbose: bool (optional, default False)
          Whether to print status data during the computation.
  
      Returns
      -------
      knn_indices: array of shape (n_samples, n_neighbors)
          The indices on the ``n_neighbors`` closest points in the dataset.
  
      knn_dists: array of shape (n_samples, n_neighbors)
          The distances to the ``n_neighbors`` closest points in the dataset.
  
      rp_forest: list of trees
          The random projection forest used for searching (if used, None otherwise)
      """
      if verbose:
          print(ts(), "Finding Nearest Neighbors")
  
      if metric == "precomputed":
          # Note that this does not support sparse distance matrices yet ...
          # Compute indices of n nearest neighbors
          knn_indices = fast_knn_indices(X, n_neighbors)
          # knn_indices = np.argsort(X)[:, :n_neighbors]
          # Compute the nearest neighbor distances
          #   (equivalent to np.sort(X)[:,:n_neighbors])
          knn_dists = X[np.arange(X.shape[0])[:, None], knn_indices].copy()
          # Prune any nearest neighbours that are infinite distance apart.
          disconnected_index = knn_dists == np.inf
          knn_indices[disconnected_index] = -1
  
          knn_search_index = None
      else:
          # TODO: Hacked values for now
          n_trees = min(64, 5 + int(round((X.shape[0]) ** 0.5 / 20.0)))
          n_iters = max(5, int(round(np.log2(X.shape[0]))))
  
          knn_search_index = NNDescent(
              X,
              n_neighbors=n_neighbors,
              metric=metric,
              metric_kwds=metric_kwds,
              random_state=random_state,
              n_trees=n_trees,
              n_iters=n_iters,
              max_candidates=60,
              low_memory=low_memory,
              n_jobs=n_jobs,
              verbose=verbose,
          )
          knn_indices, knn_dists = knn_search_index.neighbor_graph
  
      if verbose:
          print(ts(), "Finished Nearest Neighbor Search")
      return knn_indices, knn_dists, knn_search_index
  
  
  @numba.njit(
      locals={
          "knn_dists": numba.types.float32[:, ::1],
          "sigmas": numba.types.float32[::1],
          "rhos": numba.types.float32[::1],
          "val": numba.types.float32,
      },
      parallel=True,
      fastmath=True,
  )
  def compute_membership_strengths(
      knn_indices,
      knn_dists,
      sigmas,
      rhos,
      return_dists=False,
      bipartite=False,
  ):
      """Construct the membership strength data for the 1-skeleton of each local
      fuzzy simplicial set -- this is formed as a sparse matrix where each row is
      a local fuzzy simplicial set, with a membership strength for the
      1-simplex to each other data point.
  
      Parameters
      ----------
      knn_indices: array of shape (n_samples, n_neighbors)
          The indices on the ``n_neighbors`` closest points in the dataset.
  
      knn_dists: array of shape (n_samples, n_neighbors)
          The distances to the ``n_neighbors`` closest points in the dataset.
  
      sigmas: array of shape(n_samples)
          The normalization factor derived from the metric tensor approximation.
  
      rhos: array of shape(n_samples)
          The local connectivity adjustment.
  
      return_dists: bool (optional, default False)
          Whether to return the pairwise distance associated with each edge
  
      bipartite: bool (optional, default False)
          Does the nearest neighbour set represent a bipartite graph?  That is are the
          nearest neighbour indices from the same point set as the row indices?
  
      Returns
      -------
      rows: array of shape (n_samples * n_neighbors)
          Row data for the resulting sparse matrix (coo format)
  
      cols: array of shape (n_samples * n_neighbors)
          Column data for the resulting sparse matrix (coo format)
  
      vals: array of shape (n_samples * n_neighbors)
          Entries for the resulting sparse matrix (coo format)
  
      dists: array of shape (n_samples * n_neighbors)
          Distance associated with each entry in the resulting sparse matrix
      """
      n_samples = knn_indices.shape[0]
      n_neighbors = knn_indices.shape[1]
  
      rows = np.zeros(knn_indices.size, dtype=np.int32)
      cols = np.zeros(knn_indices.size, dtype=np.int32)
      vals = np.zeros(knn_indices.size, dtype=np.float32)
      if return_dists:
          dists = np.zeros(knn_indices.size, dtype=np.float32)
      else:
          dists = None
  
      for i in range(n_samples):
          for j in range(n_neighbors):
              if knn_indices[i, j] == -1:
                  continue  # We didn't get the full knn for i
              # If applied to an adjacency matrix points shouldn't be similar to themselves.
              # If applied to an incidence matrix (or bipartite) then the row and column indices are different.
              if (bipartite == False) & (knn_indices[i, j] == i):
                  val = 0.0
              elif knn_dists[i, j] - rhos[i] <= 0.0 or sigmas[i] == 0.0:
                  val = 1.0
              else:
                  val = np.exp(-((knn_dists[i, j] - rhos[i]) / (sigmas[i])))
  
              rows[i * n_neighbors + j] = i
              cols[i * n_neighbors + j] = knn_indices[i, j]
              vals[i * n_neighbors + j] = val
              if return_dists:
                  dists[i * n_neighbors + j] = knn_dists[i, j]
  
      return rows, cols, vals, dists
  
  
  def fuzzy_simplicial_set(
      X,
      n_neighbors,
      random_state,
      metric,
      metric_kwds={},
      knn_indices=None,
      knn_dists=None,
      angular=False,
      set_op_mix_ratio=1.0,
      local_connectivity=1.0,
      apply_set_operations=True,
      verbose=False,
      return_dists=None,
  ):
      """Given a set of data X, a neighborhood size, and a measure of distance
      compute the fuzzy simplicial set (here represented as a fuzzy graph in
      the form of a sparse matrix) associated to the data. This is done by
      locally approximating geodesic distance at each point, creating a fuzzy
      simplicial set for each such point, and then combining all the local
      fuzzy simplicial sets into a global one via a fuzzy union.
  
      Parameters
      ----------
      X: array of shape (n_samples, n_features)
          The data to be modelled as a fuzzy simplicial set.
  
      n_neighbors: int
          The number of neighbors to use to approximate geodesic distance.
          Larger numbers induce more global estimates of the manifold that can
          miss finer detail, while smaller values will focus on fine manifold
          structure to the detriment of the larger picture.
  
      random_state: numpy RandomState or equivalent
          A state capable being used as a numpy random state.
  
      metric: string or function (optional, default 'euclidean')
          The metric to use to compute distances in high dimensional space.
          If a string is passed it must match a valid predefined metric. If
          a general metric is required a function that takes two 1d arrays and
          returns a float can be provided. For performance purposes it is
          required that this be a numba jit'd function. Valid string metrics
          include:
              * euclidean (or l2)
              * manhattan (or l1)
              * cityblock
              * braycurtis
              * canberra
              * chebyshev
              * correlation
              * cosine
              * dice
              * hamming
              * jaccard
              * kulsinski
              * ll_dirichlet
              * mahalanobis
              * matching
              * minkowski
              * rogerstanimoto
              * russellrao
              * seuclidean
              * sokalmichener
              * sokalsneath
              * sqeuclidean
              * yule
              * wminkowski
  
          Metrics that take arguments (such as minkowski, mahalanobis etc.)
          can have arguments passed via the metric_kwds dictionary. At this
          time care must be taken and dictionary elements must be ordered
          appropriately; this will hopefully be fixed in the future.
  
      metric_kwds: dict (optional, default {})
          Arguments to pass on to the metric, such as the ``p`` value for
          Minkowski distance.
  
      knn_indices: array of shape (n_samples, n_neighbors) (optional)
          If the k-nearest neighbors of each point has already been calculated
          you can pass them in here to save computation time. This should be
          an array with the indices of the k-nearest neighbors as a row for
          each data point.
  
      knn_dists: array of shape (n_samples, n_neighbors) (optional)
          If the k-nearest neighbors of each point has already been calculated
          you can pass them in here to save computation time. This should be
          an array with the distances of the k-nearest neighbors as a row for
          each data point.
  
      angular: bool (optional, default False)
          Whether to use angular/cosine distance for the random projection
          forest for seeding NN-descent to determine approximate nearest
          neighbors.
  
      set_op_mix_ratio: float (optional, default 1.0)
          Interpolate between (fuzzy) union and intersection as the set operation
          used to combine local fuzzy simplicial sets to obtain a global fuzzy
          simplicial sets. Both fuzzy set operations use the product t-norm.
          The value of this parameter should be between 0.0 and 1.0; a value of
          1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy
          intersection.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      return_dists: bool or None (optional, default None)
          Whether to return the pairwise distance associated with each edge.
  
      Returns
      -------
      fuzzy_simplicial_set: coo_matrix
          A fuzzy simplicial set represented as a sparse matrix. The (i,
          j) entry of the matrix represents the membership strength of the
          1-simplex between the ith and jth sample points.
      """
      if knn_indices is None or knn_dists is None:
          knn_indices, knn_dists, _ = nearest_neighbors(
              X,
              n_neighbors,
              metric,
              metric_kwds,
              angular,
              random_state,
              verbose=verbose,
          )
  
      knn_dists = knn_dists.astype(np.float32)
  
      sigmas, rhos = smooth_knn_dist(
          knn_dists,
          float(n_neighbors),
          local_connectivity=float(local_connectivity),
      )
  
      rows, cols, vals, dists = compute_membership_strengths(
          knn_indices, knn_dists, sigmas, rhos, return_dists
      )
  
      result = scipy.sparse.coo_matrix(
          (vals, (rows, cols)), shape=(X.shape[0], X.shape[0])
      )
      result.eliminate_zeros()
  
      if apply_set_operations:
          transpose = result.transpose()
  
          prod_matrix = result.multiply(transpose)
  
          result = (
              set_op_mix_ratio * (result + transpose - prod_matrix)
              + (1.0 - set_op_mix_ratio) * prod_matrix
          )
  
      result.eliminate_zeros()
  
      if return_dists is None:
          return result, sigmas, rhos
      else:
          if return_dists:
              dmat = scipy.sparse.coo_matrix(
                  (dists, (rows, cols)), shape=(X.shape[0], X.shape[0])
              )
  
              dists = dmat.maximum(dmat.transpose()).todok()
          else:
              dists = None
  
          return result, sigmas, rhos, dists
  
  
  @numba.njit()
  def fast_intersection(rows, cols, values, target, unknown_dist=1.0, far_dist=5.0):
      """Under the assumption of categorical distance for the intersecting
      simplicial set perform a fast intersection.
  
      Parameters
      ----------
      rows: array
          An array of the row of each non-zero in the sparse matrix
          representation.
  
      cols: array
          An array of the column of each non-zero in the sparse matrix
          representation.
  
      values: array
          An array of the value of each non-zero in the sparse matrix
          representation.
  
      target: array of shape (n_samples)
          The categorical labels to use in the intersection.
  
      unknown_dist: float (optional, default 1.0)
          The distance an unknown label (-1) is assumed to be from any point.
  
      far_dist float (optional, default 5.0)
          The distance between unmatched labels.
  
      Returns
      -------
      None
      """
      for nz in range(rows.shape[0]):
          i = rows[nz]
          j = cols[nz]
          if (target[i] == -1) or (target[j] == -1):
              values[nz] *= np.exp(-unknown_dist)
          elif target[i] != target[j]:
              values[nz] *= np.exp(-far_dist)
  
      return
  
  
  @numba.jit()
  def fast_metric_intersection(
      rows, cols, values, discrete_space, metric, metric_args, scale
  ):
      """Under the assumption of categorical distance for the intersecting
      simplicial set perform a fast intersection.
  
      Parameters
      ----------
      rows: array
          An array of the row of each non-zero in the sparse matrix
          representation.
  
      cols: array
          An array of the column of each non-zero in the sparse matrix
          representation.
  
      values: array of shape
          An array of the values of each non-zero in the sparse matrix
          representation.
  
      discrete_space: array of shape (n_samples, n_features)
          The vectors of categorical labels to use in the intersection.
  
      metric: numba function
          The function used to calculate distance over the target array.
  
      scale: float
          A scaling to apply to the metric.
  
      Returns
      -------
      None
      """
      for nz in range(rows.shape[0]):
          i = rows[nz]
          j = cols[nz]
          dist = metric(discrete_space[i], discrete_space[j], *metric_args)
          values[nz] *= np.exp(-(scale * dist))
  
      return
  
  
  @numba.njit()
  def reprocess_row(probabilities, k=15, n_iters=32):
      target = np.log2(k)
  
      lo = 0.0
      hi = NPY_INFINITY
      mid = 1.0
  
      for n in range(n_iters):
  
          psum = 0.0
          for j in range(probabilities.shape[0]):
              psum += pow(probabilities[j], mid)
  
          if np.fabs(psum - target) < SMOOTH_K_TOLERANCE:
              break
  
          if psum < target:
              hi = mid
              mid = (lo + hi) / 2.0
          else:
              lo = mid
              if hi == NPY_INFINITY:
                  mid *= 2
              else:
                  mid = (lo + hi) / 2.0
  
      return np.power(probabilities, mid)
  
  
  @numba.njit()
  def reset_local_metrics(simplicial_set_indptr, simplicial_set_data):
      for i in range(simplicial_set_indptr.shape[0] - 1):
          simplicial_set_data[
              simplicial_set_indptr[i] : simplicial_set_indptr[i + 1]
          ] = reprocess_row(
              simplicial_set_data[simplicial_set_indptr[i] : simplicial_set_indptr[i + 1]]
          )
      return
  
  
  def reset_local_connectivity(simplicial_set, reset_local_metric=False):
      """Reset the local connectivity requirement -- each data sample should
      have complete confidence in at least one 1-simplex in the simplicial set.
      We can enforce this by locally rescaling confidences, and then remerging the
      different local simplicial sets together.
  
      Parameters
      ----------
      simplicial_set: sparse matrix
          The simplicial set for which to recalculate with respect to local
          connectivity.
  
      Returns
      -------
      simplicial_set: sparse_matrix
          The recalculated simplicial set, now with the local connectivity
          assumption restored.
      """
      simplicial_set = normalize(simplicial_set, norm="max")
      if reset_local_metric:
          simplicial_set = simplicial_set.tocsr()
          reset_local_metrics(simplicial_set.indptr, simplicial_set.data)
          simplicial_set = simplicial_set.tocoo()
      transpose = simplicial_set.transpose()
      prod_matrix = simplicial_set.multiply(transpose)
      simplicial_set = simplicial_set + transpose - prod_matrix
      simplicial_set.eliminate_zeros()
  
      return simplicial_set
  
  
  def discrete_metric_simplicial_set_intersection(
      simplicial_set,
      discrete_space,
      unknown_dist=1.0,
      far_dist=5.0,
      metric=None,
      metric_kws={},
      metric_scale=1.0,
  ):
      """Combine a fuzzy simplicial set with another fuzzy simplicial set
      generated from discrete metric data using discrete distances. The target
      data is assumed to be categorical label data (a vector of labels),
      and this will update the fuzzy simplicial set to respect that label data.
  
      TODO: optional category cardinality based weighting of distance
  
      Parameters
      ----------
      simplicial_set: sparse matrix
          The input fuzzy simplicial set.
  
      discrete_space: array of shape (n_samples)
          The categorical labels to use in the intersection.
  
      unknown_dist: float (optional, default 1.0)
          The distance an unknown label (-1) is assumed to be from any point.
  
      far_dist: float (optional, default 5.0)
          The distance between unmatched labels.
  
      metric: str (optional, default None)
          If not None, then use this metric to determine the
          distance between values.
  
      metric_scale: float (optional, default 1.0)
          If using a custom metric scale the distance values by
          this value -- this controls the weighting of the
          intersection. Larger values weight more toward target.
  
      Returns
      -------
      simplicial_set: sparse matrix
          The resulting intersected fuzzy simplicial set.
      """
      simplicial_set = simplicial_set.tocoo()
  
      if metric is not None:
          # We presume target is now a 2d array, with each row being a
          # vector of target info
          if metric in dist.named_distances:
              metric_func = dist.named_distances[metric]
          else:
              raise ValueError("Discrete intersection metric is not recognized")
  
          fast_metric_intersection(
              simplicial_set.row,
              simplicial_set.col,
              simplicial_set.data,
              discrete_space,
              metric_func,
              tuple(metric_kws.values()),
              metric_scale,
          )
      else:
          fast_intersection(
              simplicial_set.row,
              simplicial_set.col,
              simplicial_set.data,
              discrete_space,
              unknown_dist,
              far_dist,
          )
  
      simplicial_set.eliminate_zeros()
  
      return reset_local_connectivity(simplicial_set)
  
  
  def general_simplicial_set_intersection(
      simplicial_set1, simplicial_set2, weight=0.5, right_complement=False
  ):
  
      if right_complement:
          result = simplicial_set1.tocoo()
      else:
          result = (simplicial_set1 + simplicial_set2).tocoo()
      left = simplicial_set1.tocsr()
      right = simplicial_set2.tocsr()
  
      sparse.general_sset_intersection(
          left.indptr,
          left.indices,
          left.data,
          right.indptr,
          right.indices,
          right.data,
          result.row,
          result.col,
          result.data,
          mix_weight=weight,
          right_complement=right_complement,
      )
  
      return result
  
  
  def general_simplicial_set_union(simplicial_set1, simplicial_set2):
      result = (simplicial_set1 + simplicial_set2).tocoo()
      left = simplicial_set1.tocsr()
      right = simplicial_set2.tocsr()
  
      sparse.general_sset_union(
          left.indptr,
          left.indices,
          left.data,
          right.indptr,
          right.indices,
          right.data,
          result.row,
          result.col,
          result.data,
      )
  
      return result
  
  
  def make_epochs_per_sample(weights, n_epochs):
      """Given a set of weights and number of epochs generate the number of
      epochs per sample for each weight.
  
      Parameters
      ----------
      weights: array of shape (n_1_simplices)
          The weights ofhow much we wish to sample each 1-simplex.
  
      n_epochs: int
          The total number of epochs we want to train for.
  
      Returns
      -------
      An array of number of epochs per sample, one for each 1-simplex.
      """
      result = -1.0 * np.ones(weights.shape[0], dtype=np.float64)
      n_samples = n_epochs * (weights / weights.max())
      result[n_samples > 0] = float(n_epochs) / n_samples[n_samples > 0]
      return result
  
  
  def simplicial_set_embedding(
      data,
      graph,
      n_components,
      initial_alpha,
      a,
      b,
      gamma,
      negative_sample_rate,
      n_epochs,
      init,
      random_state,
      metric,
      metric_kwds,
      densmap,
      densmap_kwds,
      output_dens,
      output_metric=dist.named_distances_with_gradients["euclidean"],
      output_metric_kwds={},
      euclidean_output=True,
      parallel=False,
      verbose=False,
  ):
      """Perform a fuzzy simplicial set embedding, using a specified
      initialisation method and then minimizing the fuzzy set cross entropy
      between the 1-skeletons of the high and low dimensional fuzzy simplicial
      sets.
  
      Parameters
      ----------
      data: array of shape (n_samples, n_features)
          The source data to be embedded by UMAP.
  
      graph: sparse matrix
          The 1-skeleton of the high dimensional fuzzy simplicial set as
          represented by a graph for which we require a sparse matrix for the
          (weighted) adjacency matrix.
  
      n_components: int
          The dimensionality of the euclidean space into which to embed the data.
  
      initial_alpha: float
          Initial learning rate for the SGD.
  
      a: float
          Parameter of differentiable approximation of right adjoint functor
  
      b: float
          Parameter of differentiable approximation of right adjoint functor
  
      gamma: float
          Weight to apply to negative samples.
  
      negative_sample_rate: int (optional, default 5)
          The number of negative samples to select per positive sample
          in the optimization process. Increasing this value will result
          in greater repulsive force being applied, greater optimization
          cost, but slightly more accuracy.
  
      n_epochs: int (optional, default 0)
          The number of training epochs to be used in optimizing the
          low dimensional embedding. Larger values result in more accurate
          embeddings. If 0 is specified a value will be selected based on
          the size of the input dataset (200 for large datasets, 500 for small).
  
      init: string
          How to initialize the low dimensional embedding. Options are:
              * 'spectral': use a spectral embedding of the fuzzy 1-skeleton
              * 'random': assign initial embedding positions at random.
              * A numpy array of initial embedding positions.
  
      random_state: numpy RandomState or equivalent
          A state capable being used as a numpy random state.
  
      metric: string or callable
          The metric used to measure distance in high dimensional space; used if
          multiple connected components need to be layed out.
  
      metric_kwds: dict
          Key word arguments to be passed to the metric function; used if
          multiple connected components need to be layed out.
  
      densmap: bool
          Whether to use the density-augmented objective function to optimize
          the embedding according to the densMAP algorithm.
  
      densmap_kwds: dict
          Key word arguments to be used by the densMAP optimization.
  
      output_dens: bool
          Whether to output local radii in the original data and the embedding.
  
      output_metric: function
          Function returning the distance between two points in embedding space and
          the gradient of the distance wrt the first argument.
  
      output_metric_kwds: dict
          Key word arguments to be passed to the output_metric function.
  
      euclidean_output: bool
          Whether to use the faster code specialised for euclidean output metrics
  
      parallel: bool (optional, default False)
          Whether to run the computation using numba parallel.
          Running in parallel is non-deterministic, and is not used
          if a random seed has been set, to ensure reproducibility.
  
      verbose: bool (optional, default False)
          Whether to report information on the current progress of the algorithm.
  
      Returns
      -------
      embedding: array of shape (n_samples, n_components)
          The optimized of ``graph`` into an ``n_components`` dimensional
          euclidean space.
  
      aux_data: dict
          Auxiliary output returned with the embedding. When densMAP extension
          is turned on, this dictionary includes local radii in the original
          data (``rad_orig``) and in the embedding (``rad_emb``).
      """
      graph = graph.tocoo()
      graph.sum_duplicates()
      n_vertices = graph.shape[1]
  
      # For smaller datasets we can use more epochs
      if graph.shape[0] <= 10000:
          default_epochs = 500
      else:
          default_epochs = 200
  
      # Use more epochs for densMAP
      if densmap:
          default_epochs += 200
  
      if n_epochs is None:
          n_epochs = default_epochs
  
      if n_epochs > 10:
          graph.data[graph.data < (graph.data.max() / float(n_epochs))] = 0.0
      else:
          graph.data[graph.data < (graph.data.max() / float(default_epochs))] = 0.0
  
      graph.eliminate_zeros()
  
      if isinstance(init, str) and init == "random":
          embedding = random_state.uniform(
              low=-10.0, high=10.0, size=(graph.shape[0], n_components)
          ).astype(np.float32)
      elif isinstance(init, str) and init == "spectral":
          # We add a little noise to avoid local minima for optimization to come
          initialisation = spectral_layout(
              data,
              graph,
              n_components,
              random_state,
              metric=metric,
              metric_kwds=metric_kwds,
          )
          expansion = 10.0 / np.abs(initialisation).max()
          embedding = (initialisation * expansion).astype(
              np.float32
          ) + random_state.normal(
              scale=0.0001, size=[graph.shape[0], n_components]
          ).astype(
              np.float32
          )
      else:
          init_data = np.array(init)
          if len(init_data.shape) == 2:
              if np.unique(init_data, axis=0).shape[0] < init_data.shape[0]:
                  tree = KDTree(init_data)
                  dist, ind = tree.query(init_data, k=2)
                  nndist = np.mean(dist[:, 1])
                  embedding = init_data + random_state.normal(
                      scale=0.001 * nndist, size=init_data.shape
                  ).astype(np.float32)
              else:
                  embedding = init_data
  
      epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs)
  
      head = graph.row
      tail = graph.col
      weight = graph.data
  
      rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
      aux_data = {}
  
      if densmap or output_dens:
          if verbose:
              print(ts() + " Computing original densities")
  
          dists = densmap_kwds["graph_dists"]
  
          mu_sum = np.zeros(n_vertices, dtype=np.float32)
          ro = np.zeros(n_vertices, dtype=np.float32)
          for i in range(len(head)):
              j = head[i]
              k = tail[i]
  
              D = dists[j, k] * dists[j, k]  # match sq-Euclidean used for embedding
              mu = graph.data[i]
  
              ro[j] += mu * D
              ro[k] += mu * D
              mu_sum[j] += mu
              mu_sum[k] += mu
  
          epsilon = 1e-8
          ro = np.log(epsilon + (ro / mu_sum))
  
          if densmap:
              R = (ro - np.mean(ro)) / np.std(ro)
              densmap_kwds["mu"] = graph.data
              densmap_kwds["mu_sum"] = mu_sum
              densmap_kwds["R"] = R
  
          if output_dens:
              aux_data["rad_orig"] = ro
  
      embedding = (
          10.0
          * (embedding - np.min(embedding, 0))
          / (np.max(embedding, 0) - np.min(embedding, 0))
      ).astype(np.float32, order="C")
  
      if euclidean_output:
          embedding = optimize_layout_euclidean(
              embedding,
              embedding,
              head,
              tail,
              n_epochs,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              initial_alpha,
              negative_sample_rate,
              parallel=parallel,
              verbose=verbose,
              densmap=densmap,
              densmap_kwds=densmap_kwds,
              move_other=True,
          )
      else:
          embedding = optimize_layout_generic(
              embedding,
              embedding,
              head,
              tail,
              n_epochs,
              n_vertices,
              epochs_per_sample,
              a,
              b,
              rng_state,
              gamma,
              initial_alpha,
              negative_sample_rate,
              output_metric,
              tuple(output_metric_kwds.values()),
              verbose=verbose,
              move_other=True,
          )
      if output_dens:
          if verbose:
              print(ts() + " Computing embedding densities")
  
          # Compute graph in embedding
          (knn_indices, knn_dists, rp_forest,) = nearest_neighbors(
              embedding,
              densmap_kwds["n_neighbors"],
              "euclidean",
              {},
              False,
              random_state,
              verbose=verbose,
          )
  
          emb_graph, emb_sigmas, emb_rhos, emb_dists = fuzzy_simplicial_set(
              embedding,
              densmap_kwds["n_neighbors"],
              random_state,
              "euclidean",
              {},
              knn_indices,
              knn_dists,
              verbose=verbose,
              return_dists=True,
          )
  
          emb_graph = emb_graph.tocoo()
          emb_graph.sum_duplicates()
          emb_graph.eliminate_zeros()
  
          n_vertices = emb_graph.shape[1]
  
          mu_sum = np.zeros(n_vertices, dtype=np.float32)
          re = np.zeros(n_vertices, dtype=np.float32)
  
          head = emb_graph.row
          tail = emb_graph.col
          for i in range(len(head)):
              j = head[i]
              k = tail[i]
  
              D = emb_dists[j, k]
              mu = emb_graph.data[i]
  
              re[j] += mu * D
              re[k] += mu * D
              mu_sum[j] += mu
              mu_sum[k] += mu
  
          epsilon = 1e-8
          re = np.log(epsilon + (re / mu_sum))
  
          aux_data["rad_emb"] = re
  
      return embedding, aux_data
  
  
  @numba.njit()
  def init_transform(indices, weights, embedding):
      """Given indices and weights and an original embeddings
      initialize the positions of new points relative to the
      indices and weights (of their neighbors in the source data).
  
      Parameters
      ----------
      indices: array of shape (n_new_samples, n_neighbors)
          The indices of the neighbors of each new sample
  
      weights: array of shape (n_new_samples, n_neighbors)
          The membership strengths of associated 1-simplices
          for each of the new samples.
  
      embedding: array of shape (n_samples, dim)
          The original embedding of the source data.
  
      Returns
      -------
      new_embedding: array of shape (n_new_samples, dim)
          An initial embedding of the new sample points.
      """
      result = np.zeros((indices.shape[0], embedding.shape[1]), dtype=np.float32)
  
      for i in range(indices.shape[0]):
          for j in range(indices.shape[1]):
              for d in range(embedding.shape[1]):
                  result[i, d] += weights[i, j] * embedding[indices[i, j], d]
  
      return result
  
  
  def init_graph_transform(graph, embedding):
      """Given a bipartite graph representing the 1-simplices and strengths between the
       new points and the original data set along with an embedding of the original points
      initialize the positions of new points relative to the strengths (of their neighbors in the source data).
  
      If a point is in our original data set it embeds at the original points coordinates.
      If a point has no neighbours in our original dataset it embeds as the np.nan vector.
      Otherwise a point is the weighted average of it's neighbours embedding locations.
  
      Parameters
      ----------
      graph: csr_matrix (n_new_samples, n_samples)
          A matrix indicating the the 1-simplices and their associated strengths.  These strengths should
          be values between zero and one and not normalized.  One indicating that the new point was identical
          to one of our original points.
  
      embedding: array of shape (n_samples, dim)
          The original embedding of the source data.
  
      Returns
      -------
      new_embedding: array of shape (n_new_samples, dim)
          An initial embedding of the new sample points.
      """
      result = np.zeros((graph.shape[0], embedding.shape[1]), dtype=np.float32)
  
      for row_index in range(graph.shape[0]):
          num_neighbours = len(graph[row_index].indices)
          if num_neighbours == 0:
              result[row_index] = np.nan
              continue
          row_sum = np.sum(graph[row_index])
          for col_index in graph[row_index].indices:
              if graph[row_index, col_index] == 1:
                  result[row_index, :] = embedding[col_index, :]
                  break
              for d in range(embedding.shape[1]):
                  result[row_index, d] += (
                      graph[row_index, col_index] / row_sum * embedding[col_index, d]
                  )
  
      return result
  
  
  @numba.njit()
  def init_update(current_init, n_original_samples, indices):
      for i in range(n_original_samples, indices.shape[0]):
          n = 0
          for j in range(indices.shape[1]):
              for d in range(current_init.shape[1]):
                  if indices[i, j] < n_original_samples:
                      n += 1
                      current_init[i, d] += current_init[indices[i, j], d]
          for d in range(current_init.shape[1]):
              current_init[i, d] /= n
  
      return
  
  
  def find_ab_params(spread, min_dist):
      """Fit a, b params for the differentiable curve used in lower
      dimensional fuzzy simplicial complex construction. We want the
      smooth curve (from a pre-defined family with simple gradient) that
      best matches an offset exponential decay.
      """
  
      def curve(x, a, b):
          return 1.0 / (1.0 + a * x ** (2 * b))
  
      xv = np.linspace(0, spread * 3, 300)
      yv = np.zeros(xv.shape)
      yv[xv < min_dist] = 1.0
      yv[xv >= min_dist] = np.exp(-(xv[xv >= min_dist] - min_dist) / spread)
      params, covar = curve_fit(curve, xv, yv)
      return params[0], params[1]
  
  
  class UMAP(BaseEstimator):
      """Uniform Manifold Approximation and Projection
  
      Finds a low dimensional embedding of the data that approximates
      an underlying manifold.
  
      Parameters
      ----------
      n_neighbors: float (optional, default 15)
          The size of local neighborhood (in terms of number of neighboring
          sample points) used for manifold approximation. Larger values
          result in more global views of the manifold, while smaller
          values result in more local data being preserved. In general
          values should be in the range 2 to 100.
  
      n_components: int (optional, default 2)
          The dimension of the space to embed into. This defaults to 2 to
          provide easy visualization, but can reasonably be set to any
          integer value in the range 2 to 100.
  
      metric: string or function (optional, default 'euclidean')
          The metric to use to compute distances in high dimensional space.
          If a string is passed it must match a valid predefined metric. If
          a general metric is required a function that takes two 1d arrays and
          returns a float can be provided. For performance purposes it is
          required that this be a numba jit'd function. Valid string metrics
          include:
              * euclidean
              * manhattan
              * chebyshev
              * minkowski
              * canberra
              * braycurtis
              * mahalanobis
              * wminkowski
              * seuclidean
              * cosine
              * correlation
              * haversine
              * hamming
              * jaccard
              * dice
              * russelrao
              * kulsinski
              * ll_dirichlet
              * hellinger
              * rogerstanimoto
              * sokalmichener
              * sokalsneath
              * yule
          Metrics that take arguments (such as minkowski, mahalanobis etc.)
          can have arguments passed via the metric_kwds dictionary. At this
          time care must be taken and dictionary elements must be ordered
          appropriately; this will hopefully be fixed in the future.
  
      n_epochs: int (optional, default None)
          The number of training epochs to be used in optimizing the
          low dimensional embedding. Larger values result in more accurate
          embeddings. If None is specified a value will be selected based on
          the size of the input dataset (200 for large datasets, 500 for small).
  
      learning_rate: float (optional, default 1.0)
          The initial learning rate for the embedding optimization.
  
      init: string (optional, default 'spectral')
          How to initialize the low dimensional embedding. Options are:
              * 'spectral': use a spectral embedding of the fuzzy 1-skeleton
              * 'random': assign initial embedding positions at random.
              * A numpy array of initial embedding positions.
  
      min_dist: float (optional, default 0.1)
          The effective minimum distance between embedded points. Smaller values
          will result in a more clustered/clumped embedding where nearby points
          on the manifold are drawn closer together, while larger values will
          result on a more even dispersal of points. The value should be set
          relative to the ``spread`` value, which determines the scale at which
          embedded points will be spread out.
  
      spread: float (optional, default 1.0)
          The effective scale of embedded points. In combination with ``min_dist``
          this determines how clustered/clumped the embedded points are.
  
      low_memory: bool (optional, default True)
          For some datasets the nearest neighbor computation can consume a lot of
          memory. If you find that UMAP is failing due to memory constraints
          consider setting this option to True. This approach is more
          computationally expensive, but avoids excessive memory use.
  
      set_op_mix_ratio: float (optional, default 1.0)
          Interpolate between (fuzzy) union and intersection as the set operation
          used to combine local fuzzy simplicial sets to obtain a global fuzzy
          simplicial sets. Both fuzzy set operations use the product t-norm.
          The value of this parameter should be between 0.0 and 1.0; a value of
          1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy
          intersection.
  
      local_connectivity: int (optional, default 1)
          The local connectivity required -- i.e. the number of nearest
          neighbors that should be assumed to be connected at a local level.
          The higher this value the more connected the manifold becomes
          locally. In practice this should be not more than the local intrinsic
          dimension of the manifold.
  
      repulsion_strength: float (optional, default 1.0)
          Weighting applied to negative samples in low dimensional embedding
          optimization. Values higher than one will result in greater weight
          being given to negative samples.
  
      negative_sample_rate: int (optional, default 5)
          The number of negative samples to select per positive sample
          in the optimization process. Increasing this value will result
          in greater repulsive force being applied, greater optimization
          cost, but slightly more accuracy.
  
      transform_queue_size: float (optional, default 4.0)
          For transform operations (embedding new points using a trained model_
          this will control how aggressively to search for nearest neighbors.
          Larger values will result in slower performance but more accurate
          nearest neighbor evaluation.
  
      a: float (optional, default None)
          More specific parameters controlling the embedding. If None these
          values are set automatically as determined by ``min_dist`` and
          ``spread``.
      b: float (optional, default None)
          More specific parameters controlling the embedding. If None these
          values are set automatically as determined by ``min_dist`` and
          ``spread``.
  
      random_state: int, RandomState instance or None, optional (default: None)
          If int, random_state is the seed used by the random number generator;
          If RandomState instance, random_state is the random number generator;
          If None, the random number generator is the RandomState instance used
          by `np.random`.
  
      metric_kwds: dict (optional, default None)
          Arguments to pass on to the metric, such as the ``p`` value for
          Minkowski distance. If None then no arguments are passed on.
  
      angular_rp_forest: bool (optional, default False)
          Whether to use an angular random projection forest to initialise
          the approximate nearest neighbor search. This can be faster, but is
          mostly on useful for metric that use an angular style distance such
          as cosine, correlation etc. In the case of those metrics angular forests
          will be chosen automatically.
  
      target_n_neighbors: int (optional, default -1)
          The number of nearest neighbors to use to construct the target simplcial
          set. If set to -1 use the ``n_neighbors`` value.
  
      target_metric: string or callable (optional, default 'categorical')
          The metric used to measure distance for a target array is using supervised
          dimension reduction. By default this is 'categorical' which will measure
          distance in terms of whether categories match or are different. Furthermore,
          if semi-supervised is required target values of -1 will be trated as
          unlabelled under the 'categorical' metric. If the target array takes
          continuous values (e.g. for a regression problem) then metric of 'l1'
          or 'l2' is probably more appropriate.
  
      target_metric_kwds: dict (optional, default None)
          Keyword argument to pass to the target metric when performing
          supervised dimension reduction. If None then no arguments are passed on.
  
      target_weight: float (optional, default 0.5)
          weighting factor between data topology and target topology. A value of
          0.0 weights entirely on data, a value of 1.0 weights entirely on target.
          The default of 0.5 balances the weighting equally between data and target.
  
      transform_seed: int (optional, default 42)
          Random seed used for the stochastic aspects of the transform operation.
          This ensures consistency in transform operations.
  
      verbose: bool (optional, default False)
          Controls verbosity of logging.
  
      unique: bool (optional, default False)
          Controls if the rows of your data should be uniqued before being
          embedded.  If you have more duplicates than you have n_neighbour
          you can have the identical data points lying in different regions of
          your space.  It also violates the definition of a metric.
          For to map from internal structures back to your data use the variable
          _unique_inverse_.
  
      densmap: bool (optional, default False)
          Specifies whether the density-augmented objective of densMAP
          should be used for optimization. Turning on this option generates
          an embedding where the local densities are encouraged to be correlated
          with those in the original space. Parameters below with the prefix 'dens'
          further control the behavior of this extension.
  
      dens_lambda: float (optional, default 2.0)
          Controls the regularization weight of the density correlation term
          in densMAP. Higher values prioritize density preservation over the
          UMAP objective, and vice versa for values closer to zero. Setting this
          parameter to zero is equivalent to running the original UMAP algorithm.
  
      dens_frac: float (optional, default 0.3)
          Controls the fraction of epochs (between 0 and 1) where the
          density-augmented objective is used in densMAP. The first
          (1 - dens_frac) fraction of epochs optimize the original UMAP objective
          before introducing the density correlation term.
  
      dens_var_shift: float (optional, default 0.1)
          A small constant added to the variance of local radii in the
          embedding when calculating the density correlation objective to
          prevent numerical instability from dividing by a small number
  
      output_dens: float (optional, default False)
          Determines whether the local radii of the final embedding (an inverse
          measure of local density) are computed and returned in addition to
          the embedding. If set to True, local radii of the original data
          are also included in the output for comparison; the output is a tuple
          (embedding, original local radii, embedding local radii). This option
          can also be used when densmap=False to calculate the densities for
          UMAP embeddings.
  
      disconnection_distance: float (optional, default np.inf or maximal value for bounded distances)
          Disconnect any vertices of distance greater than or equal to disconnection_distance when approximating the
          manifold via our k-nn graph. This is particularly useful in the case that you have a bounded metric.  The
          UMAP assumption that we have a connected manifold can be problematic when you have points that are maximally
          different from all the rest of your data.  The connected manifold assumption will make such points have perfect
          similarity to a random set of other points.  Too many such points will artificially connect your space.
      """
  
      def __init__(
          self,
          n_neighbors=15,
          n_components=2,
          metric="euclidean",
          metric_kwds=None,
          output_metric="euclidean",
          output_metric_kwds=None,
          n_epochs=None,
          learning_rate=1.0,
          init="spectral",
          min_dist=0.1,
          spread=1.0,
          low_memory=True,
          n_jobs=-1,
          set_op_mix_ratio=1.0,
          local_connectivity=1.0,
          repulsion_strength=1.0,
          negative_sample_rate=5,
          transform_queue_size=4.0,
          a=None,
          b=None,
          random_state=None,
          angular_rp_forest=False,
          target_n_neighbors=-1,
          target_metric="categorical",
          target_metric_kwds=None,
          target_weight=0.5,
          transform_seed=42,
          transform_mode="embedding",
          force_approximation_algorithm=False,
          verbose=False,
          unique=False,
          densmap=False,
          dens_lambda=2.0,
          dens_frac=0.3,
          dens_var_shift=0.1,
          output_dens=False,
          disconnection_distance=None,
      ):
          self.n_neighbors = n_neighbors
          self.metric = metric
          self.output_metric = output_metric
          self.target_metric = target_metric
          self.metric_kwds = metric_kwds
          self.output_metric_kwds = output_metric_kwds
          self.n_epochs = n_epochs
          self.init = init
          self.n_components = n_components
          self.repulsion_strength = repulsion_strength
          self.learning_rate = learning_rate
  
          self.spread = spread
          self.min_dist = min_dist
          self.low_memory = low_memory
          self.set_op_mix_ratio = set_op_mix_ratio
          self.local_connectivity = local_connectivity
          self.negative_sample_rate = negative_sample_rate
          self.random_state = random_state
          self.angular_rp_forest = angular_rp_forest
          self.transform_queue_size = transform_queue_size
          self.target_n_neighbors = target_n_neighbors
          self.target_metric = target_metric
          self.target_metric_kwds = target_metric_kwds
          self.target_weight = target_weight
          self.transform_seed = transform_seed
          self.transform_mode = transform_mode
          self.force_approximation_algorithm = force_approximation_algorithm
          self.verbose = verbose
          self.unique = unique
  
          self.densmap = densmap
          self.dens_lambda = dens_lambda
          self.dens_frac = dens_frac
          self.dens_var_shift = dens_var_shift
          self.output_dens = output_dens
          self.disconnection_distance = disconnection_distance
  
          self.n_jobs = n_jobs
  
          self.a = a
          self.b = b
  
      def _validate_parameters(self):
          if self.set_op_mix_ratio < 0.0 or self.set_op_mix_ratio > 1.0:
              raise ValueError("set_op_mix_ratio must be between 0.0 and 1.0")
          if self.repulsion_strength < 0.0:
              raise ValueError("repulsion_strength cannot be negative")
          if self.min_dist > self.spread:
              raise ValueError("min_dist must be less than or equal to spread")
          if self.min_dist < 0.0:
              raise ValueError("min_dist cannot be negative")
          if not isinstance(self.init, str) and not isinstance(self.init, np.ndarray):
              raise ValueError("init must be a string or ndarray")
          if isinstance(self.init, str) and self.init not in (
              "spectral",
              "random",
          ):
              raise ValueError('string init values must be "spectral" or "random"')
          if (
              isinstance(self.init, np.ndarray)
              and self.init.shape[1] != self.n_components
          ):
              raise ValueError("init ndarray must match n_components value")
          if not isinstance(self.metric, str) and not callable(self.metric):
              raise ValueError("metric must be string or callable")
          if self.negative_sample_rate < 0:
              raise ValueError("negative sample rate must be positive")
          if self._initial_alpha < 0.0:
              raise ValueError("learning_rate must be positive")
          if self.n_neighbors < 2:
              raise ValueError("n_neighbors must be greater than 1")
          if self.target_n_neighbors < 2 and self.target_n_neighbors != -1:
              raise ValueError("target_n_neighbors must be greater than 1")
          if not isinstance(self.n_components, int):
              if isinstance(self.n_components, str):
                  raise ValueError("n_components must be an int")
              if self.n_components % 1 != 0:
                  raise ValueError("n_components must be a whole number")
              try:
                  # this will convert other types of int (eg. numpy int64)
                  # to Python int
                  self.n_components = int(self.n_components)
              except ValueError:
                  raise ValueError("n_components must be an int")
          if self.n_components < 1:
              raise ValueError("n_components must be greater than 0")
          if self.n_epochs is not None and (
              self.n_epochs < 0 or not isinstance(self.n_epochs, int)
          ):
              raise ValueError("n_epochs must be a nonnegative integer")
          if self.metric_kwds is None:
              self._metric_kwds = {}
          else:
              self._metric_kwds = self.metric_kwds
          if self.output_metric_kwds is None:
              self._output_metric_kwds = {}
          else:
              self._output_metric_kwds = self.output_metric_kwds
          if self.target_metric_kwds is None:
              self._target_metric_kwds = {}
          else:
              self._target_metric_kwds = self.target_metric_kwds
          # check sparsity of data upfront to set proper _input_distance_func &
          # save repeated checks later on
          if scipy.sparse.isspmatrix_csr(self._raw_data):
              self._sparse_data = True
          else:
              self._sparse_data = False
          # set input distance metric & inverse_transform distance metric
          if callable(self.metric):
              in_returns_grad = self._check_custom_metric(
                  self.metric, self._metric_kwds, self._raw_data
              )
              if in_returns_grad:
                  _m = self.metric
  
                  @numba.njit(fastmath=True)
                  def _dist_only(x, y, *kwds):
                      return _m(x, y, *kwds)[0]
  
                  self._input_distance_func = _dist_only
                  self._inverse_distance_func = self.metric
              else:
                  self._input_distance_func = self.metric
                  self._inverse_distance_func = None
                  warn(
                      "custom distance metric does not return gradient; inverse_transform will be unavailable. "
                      "To enable using inverse_transform method, define a distance function that returns a tuple "
                      "of (distance [float], gradient [np.array])"
                  )
          elif self.metric == "precomputed":
              if self.unique:
                  raise ValueError("unique is poorly defined on a precomputed metric")
              warn("using precomputed metric; inverse_transform will be unavailable")
              self._input_distance_func = self.metric
              self._inverse_distance_func = None
          elif self.metric == "hellinger" and self._raw_data.min() < 0:
              raise ValueError("Metric 'hellinger' does not support negative values")
          elif self.metric in dist.named_distances:
              if self._sparse_data:
                  if self.metric in sparse.sparse_named_distances:
                      self._input_distance_func = sparse.sparse_named_distances[
                          self.metric
                      ]
                  else:
                      raise ValueError(
                          "Metric {} is not supported for sparse data".format(self.metric)
                      )
              else:
                  self._input_distance_func = dist.named_distances[self.metric]
              try:
                  self._inverse_distance_func = dist.named_distances_with_gradients[
                      self.metric
                  ]
              except KeyError:
                  warn(
                      "gradient function is not yet implemented for {} distance metric; "
                      "inverse_transform will be unavailable".format(self.metric)
                  )
                  self._inverse_distance_func = None
          elif self.metric in pynn_named_distances:
              if self._sparse_data:
                  if self.metric in pynn_sparse_named_distances:
                      self._input_distance_func = pynn_sparse_named_distances[self.metric]
                  else:
                      raise ValueError(
                          "Metric {} is not supported for sparse data".format(self.metric)
                      )
              else:
                  self._input_distance_func = pynn_named_distances[self.metric]
  
              warn(
                  "gradient function is not yet implemented for {} distance metric; "
                  "inverse_transform will be unavailable".format(self.metric)
              )
              self._inverse_distance_func = None
          else:
              raise ValueError("metric is neither callable nor a recognised string")
          # set output distance metric
          if callable(self.output_metric):
              out_returns_grad = self._check_custom_metric(
                  self.output_metric, self._output_metric_kwds
              )
              if out_returns_grad:
                  self._output_distance_func = self.output_metric
              else:
                  raise ValueError(
                      "custom output_metric must return a tuple of (distance [float], gradient [np.array])"
                  )
          elif self.output_metric == "precomputed":
              raise ValueError("output_metric cannnot be 'precomputed'")
          elif self.output_metric in dist.named_distances_with_gradients:
              self._output_distance_func = dist.named_distances_with_gradients[
                  self.output_metric
              ]
          elif self.output_metric in dist.named_distances:
              raise ValueError(
                  "gradient function is not yet implemented for {}.".format(
                      self.output_metric
                  )
              )
          else:
              raise ValueError(
                  "output_metric is neither callable nor a recognised string"
              )
          # set angularity for NN search based on metric
          if self.metric in (
              "cosine",
              "correlation",
              "dice",
              "jaccard",
              "ll_dirichlet",
              "hellinger",
          ):
              self.angular_rp_forest = True
  
          if self.n_jobs < -1 or self.n_jobs == 0:
              raise ValueError("n_jobs must be a postive integer, or -1 (for all cores)")
  
          if self.dens_lambda < 0.0:
              raise ValueError("dens_lambda cannot be negative")
          if self.dens_frac < 0.0 or self.dens_frac > 1.0:
              raise ValueError("dens_frac must be between 0.0 and 1.0")
          if self.dens_var_shift < 0.0:
              raise ValueError("dens_var_shift cannot be negative")
  
          self._densmap_kwds = {
              "lambda": self.dens_lambda if self.densmap else 0.0,
              "frac": self.dens_frac if self.densmap else 0.0,
              "var_shift": self.dens_var_shift,
              "n_neighbors": self.n_neighbors,
          }
  
          if self.densmap:
              if self.output_metric not in ("euclidean", "l2"):
                  raise ValueError(
                      "Non-Euclidean output metric not supported for densMAP."
                  )
  
          # This will be used to prune all edges of greater than a fixed value from our knn graph.
          # We have preset defaults described in DISCONNECTION_DISTANCES for our bounded measures.
          # Otherwise a user can pass in their own value.
          if self.disconnection_distance is None:
              self._disconnection_distance = DISCONNECTION_DISTANCES.get(
                  self.metric, np.inf
              )
          elif isinstance(self.disconnection_distance, int) or isinstance(
              self.disconnection_distance, float
          ):
              self._disconnection_distance = self.disconnection_distance
          else:
              raise ValueError("disconnection_distance must either be None or a numeric.")
  
      def _check_custom_metric(self, metric, kwds, data=None):
          # quickly check to determine whether user-defined
          # self.metric/self.output_metric returns both distance and gradient
          if data is not None:
              # if checking the high-dimensional distance metric, test directly on
              # input data so we don't risk violating any assumptions potentially
              # hard-coded in the metric (e.g., bounded; non-negative)
              x, y = data[np.random.randint(0, data.shape[0], 2)]
          else:
              # if checking the manifold distance metric, simulate some data on a
              # reasonable interval with output dimensionality
              x, y = np.random.uniform(low=-10, high=10, size=(2, self.n_components))
  
          if scipy.sparse.issparse(data):
              metric_out = metric(x.indices, x.data, y.indices, y.data, **kwds)
          else:
              metric_out = metric(x, y, **kwds)
          # True if metric returns iterable of length 2, False otherwise
          return hasattr(metric_out, "__iter__") and len(metric_out) == 2
  
      def _populate_combined_params(self, *models):
          self.n_neighbors = flattened([m.n_neighbors for m in models])
          self.metric = flattened([m.metric for m in models])
          self.metric_kwds = flattened([m.metric_kwds for m in models])
          self.output_metric = flattened([m.output_metric for m in models])
  
          self.n_epochs = flattened(
              [m.n_epochs if m.n_epochs is not None else -1 for m in models]
          )
          if all([x == -1 for x in self.n_epochs]):
              self.n_epochs = None
  
          self.init = flattened([m.init for m in models])
          self.n_components = flattened([m.n_components for m in models])
          self.repulsion_strength = flattened([m.repulsion_strength for m in models])
          self.learning_rate = flattened([m.learning_rate for m in models])
  
          self.spread = flattened([m.spread for m in models])
          self.min_dist = flattened([m.min_dist for m in models])
          self.low_memory = flattened([m.low_memory for m in models])
          self.set_op_mix_ratio = flattened([m.set_op_mix_ratio for m in models])
          self.local_connectivity = flattened([m.local_connectivity for m in models])
          self.negative_sample_rate = flattened([m.negative_sample_rate for m in models])
          self.random_state = flattened([m.random_state for m in models])
          self.angular_rp_forest = flattened([m.angular_rp_forest for m in models])
          self.transform_queue_size = flattened([m.transform_queue_size for m in models])
          self.target_n_neighbors = flattened([m.target_n_neighbors for m in models])
          self.target_metric = flattened([m.target_metric for m in models])
          self.target_metric_kwds = flattened([m.target_metric_kwds for m in models])
          self.target_weight = flattened([m.target_weight for m in models])
          self.transform_seed = flattened([m.transform_seed for m in models])
          self.force_approximation_algorithm = flattened(
              [m.force_approximation_algorithm for m in models]
          )
          self.verbose = flattened([m.verbose for m in models])
          self.unique = flattened([m.unique for m in models])
  
          self.densmap = flattened([m.densmap for m in models])
          self.dens_lambda = flattened([m.dens_lambda for m in models])
          self.dens_frac = flattened([m.dens_frac for m in models])
          self.dens_var_shift = flattened([m.dens_var_shift for m in models])
          self.output_dens = flattened([m.output_dens for m in models])
  
          self.a = flattened([m.a for m in models])
          self.b = flattened([m.b for m in models])
  
          self._a = flattened([m._a for m in models])
          self._b = flattened([m._b for m in models])
  
      def __mul__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_intersection(
              self.graph_, other.graph_, 0.5
          )
          result.graph_ = reset_local_connectivity(result.graph_, True)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but multi-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def __add__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_union(self.graph_, other.graph_)
          result.graph_ = reset_local_connectivity(result.graph_, True)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but mult-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def __sub__(self, other):
  
          check_is_fitted(
              self, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
          check_is_fitted(
              other, attributes=["graph_"], msg="Only fitted UMAP models can be combined"
          )
  
          if self.graph_.shape[0] != other.graph_.shape[0]:
              raise ValueError("Only models with the equivalent samples can be combined")
  
          result = UMAP()
          result._populate_combined_params(self, other)
  
          result.graph_ = general_simplicial_set_intersection(
              self.graph_, other.graph_, weight=0.5, right_complement=True
          )
          result.graph_ = reset_local_connectivity(result.graph_, False)
  
          if scipy.sparse.csgraph.connected_components(result.graph_)[0] > 1:
              warn(
                  "Combined graph is not connected but mult-component layout is unsupported. "
                  "Falling back to random initialization."
              )
              init = "random"
          else:
              init = "spectral"
  
          result.densmap = np.any(result.densmap)
          result.output_dens = np.any(result.output_dens)
  
          result._densmap_kwds = {
              "lambda": np.max(result.dens_lambda),
              "frac": np.max(result.dens_frac),
              "var_shift": np.max(result.dens_var_shift),
              "n_neighbors": np.max(result.n_neighbors),
          }
  
          if result.n_epochs is None:
              n_epochs = -1
          else:
              n_epochs = np.max(result.n_epochs)
  
          result.embedding_, aux_data = simplicial_set_embedding(
              None,
              result.graph_,
              np.min(result.n_components),
              np.min(result.learning_rate),
              np.mean(result._a),
              np.mean(result._b),
              np.mean(result.repulsion_strength),
              np.mean(result.negative_sample_rate),
              n_epochs,
              init,
              check_random_state(42),
              "euclidean",
              {},
              result.densmap,
              result._densmap_kwds,
              result.output_dens,
              parallel=False,
              verbose=bool(np.max(result.verbose)),
          )
  
          if result.output_dens:
              result.rad_orig_ = aux_data["rad_orig"]
              result.rad_emb_ = aux_data["rad_emb"]
  
          return result
  
      def fit(self, X, y=None):
          """Fit X into an embedded space.
  
          Optionally use y for supervised dimension reduction.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features) or (n_samples, n_samples)
              If the metric is 'precomputed' X must be a square distance
              matrix. Otherwise it contains a sample per row. If the method
              is 'exact', X may be a sparse matrix of type 'csr', 'csc'
              or 'coo'.
  
          y : array, shape (n_samples)
              A target array for supervised dimension reduction. How this is
              handled is determined by parameters UMAP was instantiated with.
              The relevant attributes are ``target_metric`` and
              ``target_metric_kwds``.
          """
  
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          self._raw_data = X
  
          # Handle all the optional arguments, setting default
          if self.a is None or self.b is None:
              self._a, self._b = find_ab_params(self.spread, self.min_dist)
          else:
              self._a = self.a
              self._b = self.b
  
          if isinstance(self.init, np.ndarray):
              init = check_array(self.init, dtype=np.float32, accept_sparse=False)
          else:
              init = self.init
  
          self._initial_alpha = self.learning_rate
  
          self._validate_parameters()
  
          if self.verbose:
              print(str(self))
  
          self._original_n_threads = numba.get_num_threads()
          if self.n_jobs > 0 and self.n_jobs is not None:
              numba.set_num_threads(self.n_jobs)
  
          # Check if we should unique the data
          # We've already ensured that we aren't in the precomputed case
          if self.unique:
              # check if the matrix is dense
              if self._sparse_data:
                  # Call a sparse unique function
                  index, inverse, counts = csr_unique(X)
              else:
                  index, inverse, counts = np.unique(
                      X,
                      return_index=True,
                      return_inverse=True,
                      return_counts=True,
                      axis=0,
                  )[1:4]
              if self.verbose:
                  print(
                      "Unique=True -> Number of data points reduced from ",
                      X.shape[0],
                      " to ",
                      X[index].shape[0],
                  )
                  most_common = np.argmax(counts)
                  print(
                      "Most common duplicate is",
                      index[most_common],
                      " with a count of ",
                      counts[most_common],
                  )
              # We'll expose an inverse map when unique=True for users to map from our internal structures to their data
              self._unique_inverse_ = inverse
          # If we aren't asking for unique use the full index.
          # This will save special cases later.
          else:
              index = list(range(X.shape[0]))
              inverse = list(range(X.shape[0]))
  
          # Error check n_neighbors based on data size
          if X[index].shape[0] <= self.n_neighbors:
              if X[index].shape[0] == 1:
                  self.embedding_ = np.zeros(
                      (1, self.n_components)
                  )  # needed to sklearn comparability
                  return self
  
              warn(
                  "n_neighbors is larger than the dataset size; truncating to "
                  "X.shape[0] - 1"
              )
              self._n_neighbors = X[index].shape[0] - 1
              if self.densmap:
                  self._densmap_kwds["n_neighbors"] = self._n_neighbors
          else:
              self._n_neighbors = self.n_neighbors
  
          # Note: unless it causes issues for setting 'index', could move this to
          # initial sparsity check above
          if self._sparse_data and not X.has_sorted_indices:
              X.sort_indices()
  
          random_state = check_random_state(self.random_state)
  
          if self.verbose:
              print("Construct fuzzy simplicial set")
  
          if self.metric == "precomputed" and self._sparse_data:
              # For sparse precomputed distance matrices, we just argsort the rows to find
              # nearest neighbors. To make this easier, we expect matrices that are
              # symmetrical (so we can find neighbors by looking at rows in isolation,
              # rather than also having to consider that sample's column too).
              # print("Computing KNNs for sparse precomputed distances...")
              if sparse_tril(X).getnnz() != sparse_triu(X).getnnz():
                  raise ValueError(
                      "Sparse precomputed distance matrices should be symmetrical!"
                  )
              if not np.all(X.diagonal() == 0):
                  raise ValueError("Non-zero distances from samples to themselves!")
              self._knn_indices = np.zeros((X.shape[0], self.n_neighbors), dtype=np.int)
              self._knn_dists = np.zeros(self._knn_indices.shape, dtype=np.float)
              for row_id in range(X.shape[0]):
                  # Find KNNs row-by-row
                  row_data = X[row_id].data
                  row_indices = X[row_id].indices
                  if len(row_data) < self._n_neighbors:
                      raise ValueError(
                          "Some rows contain fewer than n_neighbors distances!"
                      )
                  row_nn_data_indices = np.argsort(row_data)[: self._n_neighbors]
                  self._knn_indices[row_id] = row_indices[row_nn_data_indices]
                  self._knn_dists[row_id] = row_data[row_nn_data_indices]
  
              # Disconnect any vertices farther apart than _disconnection_distance
              disconnected_index = self._knn_dists >= self._disconnection_distance
              self._knn_indices[disconnected_index] = -1
              self._knn_dists[disconnected_index] = np.inf
              edges_removed = disconnected_index.sum()
  
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  X[index],
                  self.n_neighbors,
                  random_state,
                  "precomputed",
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
          # Handle small cases efficiently by computing all distances
          elif X[index].shape[0] < 4096 and not self.force_approximation_algorithm:
              self._small_data = True
              try:
                  # sklearn pairwise_distances fails for callable metric on sparse data
                  _m = self.metric if self._sparse_data else self._input_distance_func
                  dmat = pairwise_distances(X[index], metric=_m, **self._metric_kwds)
              except (ValueError, TypeError) as e:
                  # metric is numba.jit'd or not supported by sklearn,
                  # fallback to pairwise special
  
                  if self._sparse_data:
                      # Get a fresh metric since we are casting to dense
                      if not callable(self.metric):
                          _m = dist.named_distances[self.metric]
                          dmat = dist.pairwise_special_metric(
                              X[index].toarray(),
                              metric=_m,
                              kwds=self._metric_kwds,
                          )
                      else:
                          dmat = dist.pairwise_special_metric(
                              X[index],
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  else:
                      dmat = dist.pairwise_special_metric(
                          X[index],
                          metric=self._input_distance_func,
                          kwds=self._metric_kwds,
                      )
              # set any values greater than disconnection_distance to be np.inf.
              # This will have no effect when _disconnection_distance is not set since it defaults to np.inf.
              edges_removed = np.sum(dmat >= self._disconnection_distance)
              dmat[dmat >= self._disconnection_distance] = np.inf
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  dmat,
                  self._n_neighbors,
                  random_state,
                  "precomputed",
                  self._metric_kwds,
                  None,
                  None,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
          else:
              # Standard case
              self._small_data = False
              # Standard case
              if self._sparse_data and self.metric in pynn_sparse_named_distances:
                  nn_metric = self.metric
              elif not self._sparse_data and self.metric in pynn_named_distances:
                  nn_metric = self.metric
              else:
                  nn_metric = self._input_distance_func
  
              (
                  self._knn_indices,
                  self._knn_dists,
                  self._knn_search_index,
              ) = nearest_neighbors(
                  X[index],
                  self._n_neighbors,
                  nn_metric,
                  self._metric_kwds,
                  self.angular_rp_forest,
                  random_state,
                  self.low_memory,
                  use_pynndescent=True,
                  n_jobs=self.n_jobs,
                  verbose=self.verbose,
              )
  
              # Disconnect any vertices farther apart than _disconnection_distance
              disconnected_index = self._knn_dists >= self._disconnection_distance
              self._knn_indices[disconnected_index] = -1
              self._knn_dists[disconnected_index] = np.inf
              edges_removed = disconnected_index.sum()
  
              (
                  self.graph_,
                  self._sigmas,
                  self._rhos,
                  self.graph_dists_,
              ) = fuzzy_simplicial_set(
                  X[index],
                  self.n_neighbors,
                  random_state,
                  nn_metric,
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
                  self.densmap or self.output_dens,
              )
              # Report the number of vertices with degree 0 in our our umap.graph_
              # This ensures that they were properly disconnected.
              vertices_disconnected = np.sum(
                  np.array(self.graph_.sum(axis=1)).flatten() == 0
              )
              raise_disconnected_warning(
                  edges_removed,
                  vertices_disconnected,
                  self._disconnection_distance,
                  self._raw_data.shape[0],
                  verbose=self.verbose,
              )
  
          # Currently not checking if any duplicate points have differing labels
          # Might be worth throwing a warning...
          if y is not None:
              len_X = len(X) if not self._sparse_data else X.shape[0]
              if len_X != len(y):
                  raise ValueError(
                      "Length of x = {len_x}, length of y = {len_y}, while it must be equal.".format(
                          len_x=len_X, len_y=len(y)
                      )
                  )
              if self.target_metric == "string":
                  y_ = y[index]
              else:
                  y_ = check_array(y, ensure_2d=False)[index]
              if self.target_metric == "categorical":
                  if self.target_weight < 1.0:
                      far_dist = 2.5 * (1.0 / (1.0 - self.target_weight))
                  else:
                      far_dist = 1.0e12
                  self.graph_ = discrete_metric_simplicial_set_intersection(
                      self.graph_, y_, far_dist=far_dist
                  )
              elif self.target_metric in dist.DISCRETE_METRICS:
                  if self.target_weight < 1.0:
                      scale = 2.5 * (1.0 / (1.0 - self.target_weight))
                  else:
                      scale = 1.0e12
                  # self.graph_ = discrete_metric_simplicial_set_intersection(
                  #     self.graph_,
                  #     y_,
                  #     metric=self.target_metric,
                  #     metric_kws=self.target_metric_kwds,
                  #     metric_scale=scale
                  # )
  
                  metric_kws = dist.get_discrete_params(y_, self.target_metric)
  
                  self.graph_ = discrete_metric_simplicial_set_intersection(
                      self.graph_,
                      y_,
                      metric=self.target_metric,
                      metric_kws=metric_kws,
                      metric_scale=scale,
                  )
              else:
                  if len(y_.shape) == 1:
                      y_ = y_.reshape(-1, 1)
                  if self.target_n_neighbors == -1:
                      target_n_neighbors = self._n_neighbors
                  else:
                      target_n_neighbors = self.target_n_neighbors
  
                  # Handle the small case as precomputed as before
                  if y.shape[0] < 4096:
                      try:
                          ydmat = pairwise_distances(
                              y_, metric=self.target_metric, **self._target_metric_kwds
                          )
                      except (TypeError, ValueError):
                          ydmat = dist.pairwise_special_metric(
                              y_,
                              metric=self.target_metric,
                              kwds=self._target_metric_kwds,
                          )
  
                      (target_graph, target_sigmas, target_rhos,) = fuzzy_simplicial_set(
                          ydmat,
                          target_n_neighbors,
                          random_state,
                          "precomputed",
                          self._target_metric_kwds,
                          None,
                          None,
                          False,
                          1.0,
                          1.0,
                          False,
                      )
                  else:
                      # Standard case
                      (target_graph, target_sigmas, target_rhos,) = fuzzy_simplicial_set(
                          y_,
                          target_n_neighbors,
                          random_state,
                          self.target_metric,
                          self._target_metric_kwds,
                          None,
                          None,
                          False,
                          1.0,
                          1.0,
                          False,
                      )
                  # product = self.graph_.multiply(target_graph)
                  # # self.graph_ = 0.99 * product + 0.01 * (self.graph_ +
                  # #                                        target_graph -
                  # #                                        product)
                  # self.graph_ = product
                  self.graph_ = general_simplicial_set_intersection(
                      self.graph_, target_graph, self.target_weight
                  )
                  self.graph_ = reset_local_connectivity(self.graph_)
              self._supervised = True
          else:
              self._supervised = False
  
          if self.densmap or self.output_dens:
              self._densmap_kwds["graph_dists"] = self.graph_dists_
  
          if self.verbose:
              print(ts(), "Construct embedding")
  
          if self.transform_mode == "embedding":
              self.embedding_, aux_data = self._fit_embed_data(
                  self._raw_data[index],
                  self.n_epochs,
                  init,
                  random_state,  # JH why raw data?
              )
              # Assign any points that are fully disconnected from our manifold(s) to have embedding
              # coordinates of np.nan.  These will be filtered by our plotting functions automatically.
              # They also prevent users from being deceived a distance query to one of these points.
              # Might be worth moving this into simplicial_set_embedding or _fit_embed_data
              disconnected_vertices = np.array(self.graph_.sum(axis=1)).flatten() == 0
              if len(disconnected_vertices) > 0:
                  self.embedding_[disconnected_vertices] = np.full(
                      self.n_components, np.nan
                  )
  
              self.embedding_ = self.embedding_[inverse]
              if self.output_dens:
                  self.rad_orig_ = aux_data["rad_orig"][inverse]
                  self.rad_emb_ = aux_data["rad_emb"][inverse]
  
          if self.verbose:
              print(ts() + " Finished embedding")
  
          numba.set_num_threads(self._original_n_threads)
          self._input_hash = joblib.hash(self._raw_data)
  
          return self
  
      def _fit_embed_data(self, X, n_epochs, init, random_state):
          """A method wrapper for simplicial_set_embedding that can be
          replaced by subclasses.
          """
          return simplicial_set_embedding(
              X,
              self.graph_,
              self.n_components,
              self._initial_alpha,
              self._a,
              self._b,
              self.repulsion_strength,
              self.negative_sample_rate,
              n_epochs,
              init,
              random_state,
              self._input_distance_func,
              self._metric_kwds,
              self.densmap,
              self._densmap_kwds,
              self.output_dens,
              self._output_distance_func,
              self._output_metric_kwds,
              self.output_metric in ("euclidean", "l2"),
              self.random_state is None,
              self.verbose,
          )
  
      def fit_transform(self, X, y=None):
          """Fit X into an embedded space and return that transformed
          output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features) or (n_samples, n_samples)
              If the metric is 'precomputed' X must be a square distance
              matrix. Otherwise it contains a sample per row.
  
          y : array, shape (n_samples)
              A target array for supervised dimension reduction. How this is
              handled is determined by parameters UMAP was instantiated with.
              The relevant attributes are ``target_metric`` and
              ``target_metric_kwds``.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Embedding of the training data in low-dimensional space.
  
          or a tuple (X_new, r_orig, r_emb) if ``output_dens`` flag is set,
          which additionally includes:
  
          r_orig: array, shape (n_samples)
              Local radii of data points in the original data space (log-transformed).
  
          r_emb: array, shape (n_samples)
              Local radii of data points in the embedding (log-transformed).
          """
          self.fit(X, y)
          if self.transform_mode == "embedding":
              if self.output_dens:
                  return self.embedding_, self.rad_orig_, self.rad_emb_
              else:
                  return self.embedding_
          elif self.transform_mode == "graph":
              return self.graph_
          else:
              raise ValueError(
                  "Unrecognized transform mode {}; should be one of 'embedding' or 'graph'".format(
                      self.transform_mode
                  )
              )
  
      def transform(self, X):
          """Transform X into the existing embedded space and return that
          transformed output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_features)
              New data to be transformed.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_components)
              Embedding of the new data in low-dimensional space.
          """
          # If we fit just a single instance then error
          if self._raw_data.shape[0] == 1:
              raise ValueError(
                  "Transform unavailable when model was fit with only a single data sample."
              )
          # If we just have the original input then short circuit things
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          x_hash = joblib.hash(X)
          if x_hash == self._input_hash:
              if self.transform_mode == "embedding":
                  return self.embedding_
              elif self.transform_mode == "graph":
                  return self.graph_
              else:
                  raise ValueError(
                      "Unrecognized transform mode {}; should be one of 'embedding' or 'graph'".format(
                          self.transform_mode
                      )
                  )
          if self.densmap:
              raise NotImplementedError(
                  "Transforming data into an existing embedding not supported for densMAP."
              )
  
          # X = check_array(X, dtype=np.float32, order="C", accept_sparse="csr")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          if self.metric == "precomputed":
              warn(
                  "Transforming new data with precomputed metric. "
                  "We are assuming the input data is a matrix of distances from the new points "
                  "to the points in the training set. If the input matrix is sparse, it should "
                  "contain distances from the new points to their nearest neighbours "
                  "or approximate nearest neighbours in the training set."
              )
              assert X.shape[1] == self._raw_data.shape[0]
              if scipy.sparse.issparse(X):
                  indices = np.full(
                      (X.shape[0], self._n_neighbors), dtype=np.int32, fill_value=-1
                  )
                  dists = np.full_like(indices, dtype=np.float32, fill_value=-1)
                  for i in range(X.shape[0]):
                      data_indices = np.argsort(X[i].data)
                      if len(data_indices) < self._n_neighbors:
                          raise ValueError(
                              f"Need at least n_neighbors ({self.n_neighbors}) distances for each row!"
                          )
                      indices[i] = X[i].indices[data_indices[: self._n_neighbors]]
                      dists[i] = X[i].data[data_indices[: self._n_neighbors]]
              else:
                  indices = np.argsort(X, axis=1)[:, : self._n_neighbors].astype(np.int32)
                  dists = np.take_along_axis(X, indices, axis=1)
              assert np.min(indices) >= 0 and np.min(dists) >= 0.0
          elif self._small_data:
              try:
                  # sklearn pairwise_distances fails for callable metric on sparse data
                  _m = self.metric if self._sparse_data else self._input_distance_func
                  dmat = pairwise_distances(
                      X, self._raw_data, metric=_m, **self._metric_kwds
                  )
              except (TypeError, ValueError):
                  # metric is numba.jit'd or not supported by sklearn,
                  # fallback to pairwise special
                  if self._sparse_data:
                      # Get a fresh metric since we are casting to dense
                      if not callable(self.metric):
                          _m = dist.named_distances[self.metric]
                          dmat = dist.pairwise_special_metric(
                              X.toarray(),
                              self._raw_data.toarray(),
                              metric=_m,
                              kwds=self._metric_kwds,
                          )
                      else:
                          dmat = dist.pairwise_special_metric(
                              X,
                              self._raw_data,
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  else:
                      dmat = dist.pairwise_special_metric(
                          X,
                          self._raw_data,
                          metric=self._input_distance_func,
                          kwds=self._metric_kwds,
                      )
              indices = np.argpartition(dmat, self._n_neighbors)[:, : self._n_neighbors]
              dmat_shortened = submatrix(dmat, indices, self._n_neighbors)
              indices_sorted = np.argsort(dmat_shortened)
              indices = submatrix(indices, indices_sorted, self._n_neighbors)
              dists = submatrix(dmat_shortened, indices_sorted, self._n_neighbors)
          else:
              epsilon = 0.24 if self._knn_search_index._angular_trees else 0.12
              indices, dists = self._knn_search_index.query(
                  X, self.n_neighbors, epsilon=epsilon
              )
  
          dists = dists.astype(np.float32, order="C")
          # Remove any nearest neighbours who's distances are greater than our disconnection_distance
          indices[dists >= self._disconnection_distance] = -1
          adjusted_local_connectivity = max(0.0, self.local_connectivity - 1.0)
          sigmas, rhos = smooth_knn_dist(
              dists,
              float(self._n_neighbors),
              local_connectivity=float(adjusted_local_connectivity),
          )
  
          rows, cols, vals, dists = compute_membership_strengths(
              indices, dists, sigmas, rhos, bipartite=True
          )
  
          graph = scipy.sparse.coo_matrix(
              (vals, (rows, cols)), shape=(X.shape[0], self._raw_data.shape[0])
          )
  
          if self.transform_mode == "graph":
              return graph
  
          # This was a very specially constructed graph with constant degree.
          # That lets us do fancy unpacking by reshaping the csr matrix indices
          # and data. Doing so relies on the constant degree assumption!
          # csr_graph = normalize(graph.tocsr(), norm="l1")
          # inds = csr_graph.indices.reshape(X.shape[0], self._n_neighbors)
          # weights = csr_graph.data.reshape(X.shape[0], self._n_neighbors)
          # embedding = init_transform(inds, weights, self.embedding_)
          # This is less fast code than the above numba.jit'd code.
          # It handles the fact that our nearest neighbour graph can now contain variable numbers of vertices.
          csr_graph = graph.tocsr()
          csr_graph.eliminate_zeros()
          embedding = init_graph_transform(csr_graph, self.embedding_)
  
          if self.n_epochs is None:
              # For smaller datasets we can use more epochs
              if graph.shape[0] <= 10000:
                  n_epochs = 100
              else:
                  n_epochs = 30
          else:
              n_epochs = int(self.n_epochs // 3.0)
  
          graph.data[graph.data < (graph.data.max() / float(n_epochs))] = 0.0
          graph.eliminate_zeros()
  
          epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs)
  
          head = graph.row
          tail = graph.col
          weight = graph.data
  
          # optimize_layout = make_optimize_layout(
          #     self._output_distance_func,
          #     tuple(self.output_metric_kwds.values()),
          # )
  
          if self.output_metric == "euclidean":
              embedding = optimize_layout_euclidean(
                  embedding,
                  self.embedding_.astype(np.float32, copy=True),  # Fixes #179 & #217,
                  head,
                  tail,
                  n_epochs,
                  graph.shape[1],
                  epochs_per_sample,
                  self._a,
                  self._b,
                  rng_state,
                  self.repulsion_strength,
                  self._initial_alpha / 4.0,
                  self.negative_sample_rate,
                  self.random_state is None,
                  verbose=self.verbose,
              )
          else:
              embedding = optimize_layout_generic(
                  embedding,
                  self.embedding_.astype(np.float32, copy=True),  # Fixes #179 & #217
                  head,
                  tail,
                  n_epochs,
                  graph.shape[1],
                  epochs_per_sample,
                  self._a,
                  self._b,
                  rng_state,
                  self.repulsion_strength,
                  self._initial_alpha / 4.0,
                  self.negative_sample_rate,
                  self._output_distance_func,
                  tuple(self._output_metric_kwds.values()),
                  verbose=self.verbose,
              )
  
          return embedding
  
      def inverse_transform(self, X):
          """Transform X in the existing embedded space back into the input
          data space and return that transformed output.
  
          Parameters
          ----------
          X : array, shape (n_samples, n_components)
              New points to be inverse transformed.
  
          Returns
          -------
          X_new : array, shape (n_samples, n_features)
              Generated data points new data in data space.
          """
  
          if self._sparse_data:
              raise ValueError("Inverse transform not available for sparse input.")
          elif self._inverse_distance_func is None:
              raise ValueError("Inverse transform not available for given metric.")
          elif self.densmap:
              raise ValueError("Inverse transform not available for densMAP.")
          elif self.n_components >= 8:
              warn(
                  "Inverse transform works best with low dimensional embeddings."
                  " Results may be poor, or this approach to inverse transform"
                  " may fail altogether! If you need a high dimensional latent"
                  " space and inverse transform operations consider using an"
                  " autoencoder."
              )
          elif self.transform_mode == "graph":
              raise ValueError(
                  "Inverse transform not available for transform_mode = 'graph'"
              )
  
          X = check_array(X, dtype=np.float32, order="C")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          # build Delaunay complex (Does this not assume a roughly euclidean output metric)?
          deltri = scipy.spatial.Delaunay(
              self.embedding_, incremental=True, qhull_options="QJ"
          )
          neighbors = deltri.simplices[deltri.find_simplex(X)]
          adjmat = scipy.sparse.lil_matrix(
              (self.embedding_.shape[0], self.embedding_.shape[0]), dtype=int
          )
          for i in np.arange(0, deltri.simplices.shape[0]):
              for j in deltri.simplices[i]:
                  if j < self.embedding_.shape[0]:
                      idx = deltri.simplices[i][
                          deltri.simplices[i] < self.embedding_.shape[0]
                      ]
                      adjmat[j, idx] = 1
                      adjmat[idx, j] = 1
  
          adjmat = scipy.sparse.csr_matrix(adjmat)
  
          min_vertices = min(self._raw_data.shape[-1], self._raw_data.shape[0])
  
          neighborhood = [
              breadth_first_search(adjmat, v[0], min_vertices=min_vertices)
              for v in neighbors
          ]
          if callable(self.output_metric):
              # need to create another numba.jit-able wrapper for callable
              # output_metrics that return a tuple (already checked that it does
              # during param validation in `fit` method)
              _out_m = self.output_metric
  
              @numba.njit(fastmath=True)
              def _output_dist_only(x, y, *kwds):
                  return _out_m(x, y, *kwds)[0]
  
              dist_only_func = _output_dist_only
          elif self.output_metric in dist.named_distances.keys():
              dist_only_func = dist.named_distances[self.output_metric]
          else:
              # shouldn't really ever get here because of checks already performed,
              # but works as a failsafe in case attr was altered manually after fitting
              raise ValueError(
                  "Unrecognized output metric: {}".format(self.output_metric)
              )
  
          dist_args = tuple(self._output_metric_kwds.values())
          distances = [
              np.array(
                  [
                      dist_only_func(X[i], self.embedding_[nb], *dist_args)
                      for nb in neighborhood[i]
                  ]
              )
              for i in range(X.shape[0])
          ]
          idx = np.array([np.argsort(e)[:min_vertices] for e in distances])
  
          dists_output_space = np.array(
              [distances[i][idx[i]] for i in range(len(distances))]
          )
          indices = np.array([neighborhood[i][idx[i]] for i in range(len(neighborhood))])
  
          rows, cols, distances = np.array(
              [
                  [i, indices[i, j], dists_output_space[i, j]]
                  for i in range(indices.shape[0])
                  for j in range(min_vertices)
              ]
          ).T
  
          # calculate membership strength of each edge
          weights = 1 / (1 + self._a * distances ** (2 * self._b))
  
          # compute 1-skeleton
          # convert 1-skeleton into coo_matrix adjacency matrix
          graph = scipy.sparse.coo_matrix(
              (weights, (rows, cols)), shape=(X.shape[0], self._raw_data.shape[0])
          )
  
          # That lets us do fancy unpacking by reshaping the csr matrix indices
          # and data. Doing so relies on the constant degree assumption!
          # csr_graph = graph.tocsr()
          csr_graph = normalize(graph.tocsr(), norm="l1")
          inds = csr_graph.indices.reshape(X.shape[0], min_vertices)
          weights = csr_graph.data.reshape(X.shape[0], min_vertices)
          inv_transformed_points = init_transform(inds, weights, self._raw_data)
  
          if self.n_epochs is None:
              # For smaller datasets we can use more epochs
              if graph.shape[0] <= 10000:
                  n_epochs = 100
              else:
                  n_epochs = 30
          else:
              n_epochs = int(self.n_epochs // 3.0)
  
          # graph.data[graph.data < (graph.data.max() / float(n_epochs))] = 0.0
          # graph.eliminate_zeros()
  
          epochs_per_sample = make_epochs_per_sample(graph.data, n_epochs)
  
          head = graph.row
          tail = graph.col
          weight = graph.data
  
          inv_transformed_points = optimize_layout_inverse(
              inv_transformed_points,
              self._raw_data,
              head,
              tail,
              weight,
              self._sigmas,
              self._rhos,
              n_epochs,
              graph.shape[1],
              epochs_per_sample,
              self._a,
              self._b,
              rng_state,
              self.repulsion_strength,
              self._initial_alpha / 4.0,
              self.negative_sample_rate,
              self._inverse_distance_func,
              tuple(self._metric_kwds.values()),
              verbose=self.verbose,
          )
  
          return inv_transformed_points
  
      def update(self, X):
          X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
          random_state = check_random_state(self.transform_seed)
          rng_state = random_state.randint(INT32_MIN, INT32_MAX, 3).astype(np.int64)
  
          original_size = self._raw_data.shape[0]
  
          if self.metric == "precomputed":
              raise ValueError("Update does not currently support precomputed metrics")
          if self._supervised:
              raise ValueError("Updating supervised models is not currently " "supported")
  
          if self._small_data:
  
              if self._sparse_data:
                  self._raw_data = scipy.sparse.vstack([self._raw_data, X])
              else:
                  self._raw_data = np.vstack([self._raw_data, X])
  
              if self._raw_data.shape[0] < 4096:
                  # still small data
                  try:
                      # sklearn pairwise_distances fails for callable metric on sparse data
                      _m = self.metric if self._sparse_data else self._input_distance_func
                      dmat = pairwise_distances(
                          self._raw_data, metric=_m, **self._metric_kwds
                      )
                  except (ValueError, TypeError) as e:
                      # metric is numba.jit'd or not supported by sklearn,
                      # fallback to pairwise special
  
                      if self._sparse_data:
                          # Get a fresh metric since we are casting to dense
                          if not callable(self.metric):
                              _m = dist.named_distances[self.metric]
                              dmat = dist.pairwise_special_metric(
                                  self._raw_data.toarray(),
                                  metric=_m,
                                  kwds=self._metric_kwds,
                              )
                          else:
                              dmat = dist.pairwise_special_metric(
                                  self._raw_data,
                                  metric=self._input_distance_func,
                                  kwds=self._metric_kwds,
                              )
                      else:
                          dmat = dist.pairwise_special_metric(
                              self._raw_data,
                              metric=self._input_distance_func,
                              kwds=self._metric_kwds,
                          )
                  self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                      dmat,
                      self._n_neighbors,
                      random_state,
                      "precomputed",
                      self._metric_kwds,
                      None,
                      None,
                      self.angular_rp_forest,
                      self.set_op_mix_ratio,
                      self.local_connectivity,
                      True,
                      self.verbose,
                  )
                  knn_indices = np.argsort(dmat)[:, : self.n_neighbors]
              else:
                  # now large data
                  self._small_data = False
                  if self._sparse_data and self.metric in pynn_sparse_named_distances:
                      nn_metric = self.metric
                  elif not self._sparse_data and self.metric in pynn_named_distances:
                      nn_metric = self.metric
                  else:
                      nn_metric = self._input_distance_func
  
                  (
                      self._knn_indices,
                      self._knn_dists,
                      self._knn_search_index,
                  ) = nearest_neighbors(
                      self._raw_data,
                      self._n_neighbors,
                      nn_metric,
                      self._metric_kwds,
                      self.angular_rp_forest,
                      random_state,
                      self.low_memory,
                      use_pynndescent=True,
                      n_jobs=self.n_jobs,
                      verbose=self.verbose,
                  )
  
                  self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                      self._raw_data,
                      self.n_neighbors,
                      random_state,
                      nn_metric,
                      self._metric_kwds,
                      self._knn_indices,
                      self._knn_dists,
                      self.angular_rp_forest,
                      self.set_op_mix_ratio,
                      self.local_connectivity,
                      True,
                      self.verbose,
                  )
                  knn_indices = self._knn_indices
  
              init = np.zeros(
                  (self._raw_data.shape[0], self.n_components), dtype=np.float32
              )
              init[:original_size] = self.embedding_
  
              init_update(init, original_size, knn_indices)
              if self.n_epochs is None:
                  n_epochs = 0
              else:
                  n_epochs = self.n_epochs
  
              self.embedding_, aux_data = simplicial_set_embedding(
                  self._raw_data,
                  self.graph_,
                  self.n_components,
                  self._initial_alpha,
                  self._a,
                  self._b,
                  self.repulsion_strength,
                  self.negative_sample_rate,
                  n_epochs,
                  init,
                  random_state,
                  self._input_distance_func,
                  self._metric_kwds,
                  self.densmap,
                  self._densmap_kwds,
                  self.output_dens,
                  self._output_distance_func,
                  self._output_metric_kwds,
                  self.output_metric in ("euclidean", "l2"),
                  self.random_state is None,
                  self.verbose,
              )
  
          else:
              self._knn_search_index.update(X)
              self._raw_data = self._knn_search_index._raw_data
              (
                  self._knn_indices,
                  self._knn_dists,
              ) = self._knn_search_index.neighbor_graph
  
              if self._sparse_data and self.metric in pynn_sparse_named_distances:
                  nn_metric = self.metric
              elif not self._sparse_data and self.metric in pynn_named_distances:
                  nn_metric = self.metric
              else:
                  nn_metric = self._input_distance_func
  
              self.graph_, self._sigmas, self._rhos = fuzzy_simplicial_set(
                  self._raw_data,
                  self.n_neighbors,
                  random_state,
                  nn_metric,
                  self._metric_kwds,
                  self._knn_indices,
                  self._knn_dists,
                  self.angular_rp_forest,
                  self.set_op_mix_ratio,
                  self.local_connectivity,
                  True,
                  self.verbose,
              )
  
              init = np.zeros(
                  (self._raw_data.shape[0], self.n_components), dtype=np.float32
              )
              init[:original_size] = self.embedding_
              init_update(init, original_size, self._knn_indices)
  
              if self.n_epochs is None:
                  n_epochs = 0
              else:
                  n_epochs = self.n_epochs
  
              self.embedding_, aux_data = simplicial_set_embedding(
                  self._raw_data,
                  self.graph_,
                  self.n_components,
                  self._initial_alpha,
                  self._a,
                  self._b,
                  self.repulsion_strength,
                  self.negative_sample_rate,
                  n_epochs,
                  init,
                  random_state,
                  self._input_distance_func,
                  self._metric_kwds,
                  self.densmap,
                  self._densmap_kwds,
                  self.output_dens,
                  self._output_distance_func,
                  self._output_metric_kwds,
                  self.output_metric in ("euclidean", "l2"),
                  self.random_state is None,
                  self.verbose,
              )
  
          if self.output_dens:
              self.rad_orig_ = aux_data["rad_orig"]
              self.rad_emb_ = aux_data["rad_emb"]
  
files: umap/umap_.py