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 :ref:`User Guide <spectral_embedding>`.
Parameters ---------- n_components : integer, default: 2 The dimension of the projected subspace.
affinity : string 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, optional, default : 1/n_features Kernel coefficient for rbf kernel.
random_state : int, RandomState instance or None, optional, default: None A pseudo random number generator used for the initialization of the lobpcg eigenvectors. 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`. Used when ``solver`` == 'amg'.
eigen_solver : None, 'arpack', 'lobpcg', or 'amg' The eigenvalue decomposition strategy to use. AMG requires pyamg to be installed. It can be faster on very large, sparse problems.
n_neighbors : int, default : max(n_samples/10 , 1) Number of nearest neighbors for nearest_neighbors graph building.
n_jobs : int or None, optional (default=None) The number of parallel jobs to run. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary <n_jobs>` for more details.
Attributes ----------
embedding_ : array, shape = (n_samples, n_components) Spectral embedding of the training matrix.
affinity_matrix_ : array, shape = (n_samples, n_samples) Affinity_matrix constructed from samples or precomputed.
n_neighbors_ : int Number of nearest neighbors effectively used.
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)
References ----------
- A Tutorial on Spectral Clustering, 2007 Ulrike von Luxburg http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
- On Spectral Clustering: Analysis and an algorithm, 2001 Andrew Y. Ng, Michael I. Jordan, Yair Weiss http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8100
- Normalized cuts and image segmentation, 2000 Jianbo Shi, Jitendra Malik http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324