sklearn.manifold.spectral_embedding

sklearn.manifold.spectral_embedding(adjacency, n_components=8, eigen_solver=None, random_state=None, eigen_tol=0.0, norm_laplacian=True, drop_first=True)

Project the sample on the first eigen vectors of the graph Laplacian.

The adjacency matrix is used to compute a normalized graph Laplacian whose spectrum (especially the eigen vectors associated to the smallest eigen values) 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 eigen vector decomposition works as expected.

Parameters:

adjacency : array-like or sparse matrix, shape: (n_samples, n_samples)

The adjacency matrix of the graph to embed.

n_components : integer, optional, default 8

The dimension of the projection subspace.

eigen_solver : {None, ‘arpack’, ‘lobpcg’, or ‘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.

random_state : int seed, RandomState instance, or None (default)

A pseudo random number generator used for the initialization of the lobpcg eigen vectors decomposition when eigen_solver == ‘amg’. By default, arpack is used.

eigen_tol : float, optional, default=0.0

Stopping criterion for eigendecomposition of the Laplacian matrix when using arpack eigen_solver.

drop_first : bool, optional, 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.

norm_laplacian : bool, optional, default=True

If True, then compute normalized Laplacian.

Returns:

embedding : array, shape=(n_samples, n_components)

The reduced samples.

Notes

Spectral embedding 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

• http://en.wikipedia.org/wiki/LOBPCG
• Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method Andrew V. Knyazev http://dx.doi.org/10.1137%2FS1064827500366124