code: def locally_linear_embedding(
      X, *, n_components=2, n_neighbors=5, metric="riemann", reg=1e-3
  ):
      """Perform a Locally Linear Embedding (LLE) of SPD matrices.
  
      As proposed in [1]_, Locally Linear Embedding (LLE) is a non-linear,
      neighborhood-preserving dimensionality reduction algorithm which consists
      of three main steps. For each point xi,
  
      1.  find its k nearest neighbors KNN(xi),
      2.  calculate the best reconstruction of xi based on its
          k-nearest neighbors (Eq.9 in [1]_),
      3.  calculate a low-dimensional embedding for all points based on
          the weights in step 2.
  
      Parameters
      ----------
      X : ndarray, shape (n_matrices, n_channels, n_channels)
          Set of SPD matrices.
      n_components : int, default=2
          Dimensionality of projected space.
      n_neighbors : int, default=5
          Number of neighbors for reconstruction of each point.
      metric : {'riemann', 'logeuclid', 'euclid'}, default: 'riemann'
          Metric used for KNN and Kernel estimation.
      reg : float, default=1e-3
          Regularization parameter.
  
      Returns
      -------
      embd : ndarray, shape (n_matrices, n_components)
          Locally linear embedding of matrices in X.
      error : float
          Error of the projected embedding.
  
      Notes
      -----
      .. versionadded:: 0.3
  
      References
      ----------
      .. [1] `Clustering and dimensionality reduction
          on Riemannian manifolds
          <https://ieeexplore.ieee.org/document/4587422>`_
          A. Goh and R. Vidal, in 2008 IEEE Conference on Computer Vision and
          Pattern Recognition
      """
      n_matrices, n_channels, n_channels = X.shape
      pairwise_distances = pairwise_distance(X, metric=metric)
      neighbors = np.array(
          [
              np.argsort(dist)[1 : n_neighbors + 1]
              for dist in pairwise_distances
          ]
      )
  
      B = barycenter_weights(X, X, neighbors, metric=metric, reg=reg)
  
      indptr = np.arange(0, n_matrices * n_neighbors + 1, n_neighbors)
      W = csr_matrix(
          (B.ravel(), neighbors.ravel(), indptr),
          shape=(n_matrices, n_matrices),
      )
      # M = (W - I).T * (W - I) = W.T * W - W.T - W + I
      # calculated in the two following lines
      M = (W.T * W - W.T - W).toarray()
      M.flat[:: M.shape[0] + 1] += 1
  
      eigen_values, eigen_vectors = eigh(
          M, subset_by_index=(1, n_components), overwrite_a=True
      )
      index = np.argsort(np.abs(eigen_values))
      embd, error = eigen_vectors[:, index], np.sum(eigen_values)
  
      return embd, error
  
imports: import numpy
  from scipy.linalg import solve, eigh
  from scipy.sparse import csr_matrix
  from sklearn.base import BaseEstimator, TransformerMixin
  from sklearn.manifold import spectral_embedding
  from .utils.kernel import kernel
  from .utils.distance import pairwise_distance
label_desc: Use image node_id 2 for calling a global function with example usage: locally_linear_embedding(X) and returns: embd,  error