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# `sklearn.manifold`.SpectralEmbedding

class sklearn.manifold.SpectralEmbedding(n_components=2, *, affinity='nearest_neighbors', gamma=None, random_state=None, eigen_solver=None, eigen_tol='auto', n_neighbors=None, n_jobs=None)

  Spectral embedding for non-linear dimensionality reduction.

  Forms an affinity matrix given by the specified function and applies spectral decomposition to the corresponding graph laplacian. The resulting transformation is given by the value of the eigenvectors for each data point.

  Note : Laplacian Eigenmaps is the actual algorithm implemented here.

  Read more in the User Guide.

  Parameters:
    n_components int, default=2

      The dimension of the projected subspace.

    affinity{‘nearest_neighbors’, ‘rbf’, ‘precomputed’, ‘precomputed_nearest_neighbors’} or callable, default=’nearest_neighbors’
      How to construct the affinity matrix.

        - ‘nearest_neighbors’ : construct the affinity matrix by computing a graph of nearest neighbors.

        - ‘rbf’ : construct the affinity matrix by computing a radial basis function (RBF) kernel.

        - ‘precomputed’ : interpret `X` as a precomputed affinity matrix.

        - ‘precomputed_nearest_neighbors’ : interpret `X` as a sparse graph of precomputed nearest neighbors, and constructs the affinity matrix by selecting the `n_neighbors` nearest neighbors.

        - callable : use passed in function as affinity the function takes in data matrix (n_samples, n_features) and return affinity matrix (n_samples, n_samples).

    gamma float, default=None

      Kernel coefficient for rbf kernel. If None, gamma will be set to 1/n_features.

    random_state int, RandomState instance or None, default=None

      A pseudo random number generator used for the initialization of the lobpcg eigen vectors decomposition when `eigen_solver == 'amg'`, and for the K-Means initialization. Use an int to make the results deterministic across calls (See Glossary).

      Note

      When using `eigen_solver == 'amg'`, it is necessary to also fix the global numpy seed with `np.random.seed(int)` to get deterministic results. See https://github.com/pyamg/pyamg/issues/139 for further information.

    eigen_solver{‘arpack’, ‘lobpcg’, ‘amg’}, default=None

      The eigenvalue decomposition strategy to use. AMG requires pyamg to be installed. It can be faster on very large, sparse problems. If None, then `'arpack'` is used.

    eigen_tol float, default=”auto”

      Stopping criterion for eigendecomposition of the Laplacian matrix. If `eigen_tol="auto"` then the passed tolerance will depend on the `eigen_solver`:

      - If `eigen_solver="arpack"`, then `eigen_tol=0.0`;

      - If `eigen_solver="lobpcg"` or `eigen_solver="amg"`, then `eigen_tol=None` which configures the underlying `lobpcg` solver to automatically resolve the value according to their heuristics. See, `scipy.sparse.linalg.lobpcg` for details.

      Note that when using `eigen_solver="lobpcg"` or `eigen_solver="amg"` values of `tol<1e-5` may lead to convergence issues and should be avoided.

      New in version 1.2.

    n_neighbors int, default=None

      Number of nearest neighbors for nearest_neighbors graph building. If None, n_neighbors will be set to max(n_samples/10, 1).

    n_jobs int, default=None

      The number of parallel jobs to run. `None` means 1 unless in a `joblib.parallel_backend` context. `-1` means using all processors. See Glossary for more details.

  Attributes:
    embedding_ndarray of shape (n_samples, n_components)

      Spectral embedding of the training matrix.

    affinity_matrix_ndarray of shape (n_samples, n_samples)

      Affinity_matrix constructed from samples or precomputed.

    n_features_in_int

      Number of features seen during fit.

      New in version 0.24.

    feature_names_in_ndarray of shape (`n_features_in_`,)

      Names of features seen during fit. Defined only when `X` has feature names that are all strings.

      New in version 1.0.

    n_neighbors_int

      Number of nearest neighbors effectively used.

  See also

  `Isomap`

    Non-linear dimensionality reduction through Isometric Mapping.

  References

  - A Tutorial on Spectral Clustering, 2007 Ulrike von Luxburg

  - On Spectral Clustering: Analysis and an algorithm, 2001 Andrew Y. Ng, Michael I. Jordan, Yair Weiss

  - Normalized cuts and image segmentation, 2000 Jianbo Shi, Jitendra Malik

  Examples

      >>> from sklearn.datasets import load_digits
      >>> from sklearn.manifold import SpectralEmbedding
      >>> X, _ = load_digits(return_X_y=True)
      >>> X.shape
      (1797, 64)
      >>> embedding = SpectralEmbedding(n_components=2)
      >>> X_transformed = embedding.fit_transform(X[:100])
      >>> X_transformed.shape
      (100, 2)

  Methods

  `fit`(X[, y]) | Fit the model from data in X.
  `fit_transform`(X[, y]) | Fit the model from data in X and transform X.
  `get_metadata_routing`() | Get metadata routing of this object.
  `get_params`([deep]) | Get parameters for this estimator.
  `set_params`(**params) | Set the parameters of this estimator.

  fit(X, y=None)

    Fit the model from data in X.

    Parameters:
      X{array-like, sparse matrix} of shape (n_samples, n_features)

        Training vector, where `n_samples` is the number of samples and `n_features` is the number of features.

        If affinity is “precomputed” X : {array-like, sparse matrix}, shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples.

      y Ignored

        Not used, present for API consistency by convention.

    Returns:
      self object

        Returns the instance itself.

  fit_transform(X, y=None)

    Fit the model from data in X and transform X.

    Parameters:
      X{array-like, sparse matrix} of shape (n_samples, n_features)

        Training vector, where `n_samples` is the number of samples and `n_features` is the number of features.

        If affinity is “precomputed” X : {array-like, sparse matrix} of shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples.

      y Ignored

        Not used, present for API consistency by convention.

    Returns:
      X_new array-like of shape (n_samples, n_components)

        Spectral embedding of the training matrix.

  get_metadata_routing()

    Get metadata routing of this object.

    Please check User Guide on how the routing mechanism works.

    Returns:
      routing MetadataRequest

        A `MetadataRequest` encapsulating routing information.

  get_params(deep=True)

    Get parameters for this estimator.

    Parameters:
      deep bool, default=True

        If True, will return the parameters for this estimator and contained subobjects that are estimators.

    Returns:
      params dict

        Parameter names mapped to their values.

  set_params(**params)

    Set the parameters of this estimator.

    The method works on simple estimators as well as on nested objects (such as `Pipeline`). The latter have parameters of the form `<component>__<parameter>` so that it’s possible to update each component of a nested object.

    Parameters:
      **params dict

        Estimator parameters.

    Returns:
      self estimator instance

        Estimator instance.

## Examples using `sklearn.manifold.SpectralEmbedding`

Various Agglomerative Clustering on a 2D embedding of digits

Various Agglomerative Clustering on a 2D embedding of digits

Comparison of Manifold Learning methods

Comparison of Manifold Learning methods

Manifold Learning methods on a severed sphere

Manifold Learning methods on a severed sphere

Manifold learning on handwritten digits: Locally Linear Embedding, Isomap…

Manifold learning on handwritten digits: Locally Linear Embedding, Isomap...