spectral_embedding

sklearn.manifold.spectral_embedding(adjacency, *, n_components=8, eigen_solver=None, random_state=None, eigen_tol='auto', norm_laplacian=True, drop_first=True)

Project the sample on the first eigenvectors of the graph Laplacian.

The adjacency matrix is used to compute a normalized graph Laplacian whose spectrum (especially the eigenvectors associated to the smallest eigenvalues) has an interpretation in terms of minimal number of cuts necessary to split the graph into comparably sized components.

This embedding can also ‘work’ even if the adjacency variable is not strictly the adjacency matrix of a graph but more generally an affinity or similarity matrix between samples (for instance the heat kernel of a euclidean distance matrix or a k-NN matrix).

However care must taken to always make the affinity matrix symmetric so that the eigenvector decomposition works as expected.

Note : Laplacian Eigenmaps is the actual algorithm implemented here.

Read more in the User Guide.

Parameters:

adjacency{array-like, sparse graph} of shape (n_samples, n_samples)

The adjacency matrix of the graph to embed.

n_componentsint, default=8

The dimension of the projection subspace.

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, but may also lead to instabilities. If None, then 'arpack' is used.

random_stateint, 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 pyamg/pyamg#139 for further information.

eigen_tolfloat, 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="amg" values of tol<1e-5 may lead to convergence issues and should be avoided.

Added in version 1.2: Added ‘auto’ option.

norm_laplacianbool, default=True

If True, then compute symmetric normalized Laplacian.

drop_firstbool, default=True

Whether to drop the first eigenvector. For spectral embedding, this should be True as the first eigenvector should be constant vector for connected graph, but for spectral clustering, this should be kept as False to retain the first eigenvector.

Returns:

embeddingndarray of shape (n_samples, n_components)

The reduced samples.

Notes

Spectral Embedding (Laplacian Eigenmaps) is most useful when the graph has one connected component. If there graph has many components, the first few eigenvectors will simply uncover the connected components of the graph.

References

• https://en.wikipedia.org/wiki/LOBPCG

• “Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method”, Andrew V. Knyazev

Examples

>>> from sklearn.datasets import load_digits
>>> from sklearn.neighbors import kneighbors_graph
>>> from sklearn.manifold import spectral_embedding
>>> X, _ = load_digits(return_X_y=True)
>>> X = X[:100]
>>> affinity_matrix = kneighbors_graph(
...     X, n_neighbors=int(X.shape[0] / 10), include_self=True
... )
>>> # make the matrix symmetric
>>> affinity_matrix = 0.5 * (affinity_matrix + affinity_matrix.T)
>>> embedding = spectral_embedding(affinity_matrix, n_components=2, random_state=42)
>>> embedding.shape
(100, 2)