"""Spectral Embedding.""" # Author: Gael Varoquaux # Wei LI # License: BSD 3 clause import warnings import numpy as np from scipy import sparse from scipy.linalg import eigh from scipy.sparse.linalg import eigsh from scipy.sparse.csgraph import connected_components from scipy.sparse.csgraph import laplacian as csgraph_laplacian from ..base import BaseEstimator from ..utils import ( check_array, check_random_state, check_symmetric, ) from ..utils._arpack import _init_arpack_v0 from ..utils.extmath import _deterministic_vector_sign_flip from ..utils.fixes import lobpcg from ..metrics.pairwise import rbf_kernel from ..neighbors import kneighbors_graph, NearestNeighbors from ..utils.validation import _deprecate_positional_args from ..utils.deprecation import deprecated def _graph_connected_component(graph, node_id): """Find the largest graph connected components that contains one given node. Parameters ---------- graph : array-like of shape (n_samples, n_samples) Adjacency matrix of the graph, non-zero weight means an edge between the nodes. node_id : int The index of the query node of the graph. Returns ------- connected_components_matrix : array-like of shape (n_samples,) An array of bool value indicating the indexes of the nodes belonging to the largest connected components of the given query node. """ n_node = graph.shape[0] if sparse.issparse(graph): # speed up row-wise access to boolean connection mask graph = graph.tocsr() connected_nodes = np.zeros(n_node, dtype=bool) nodes_to_explore = np.zeros(n_node, dtype=bool) nodes_to_explore[node_id] = True for _ in range(n_node): last_num_component = connected_nodes.sum() np.logical_or(connected_nodes, nodes_to_explore, out=connected_nodes) if last_num_component >= connected_nodes.sum(): break indices = np.where(nodes_to_explore)[0] nodes_to_explore.fill(False) for i in indices: if sparse.issparse(graph): neighbors = graph[i].toarray().ravel() else: neighbors = graph[i] np.logical_or(nodes_to_explore, neighbors, out=nodes_to_explore) return connected_nodes def _graph_is_connected(graph): """ Return whether the graph is connected (True) or Not (False). Parameters ---------- graph : {array-like, sparse matrix} of shape (n_samples, n_samples) Adjacency matrix of the graph, non-zero weight means an edge between the nodes. Returns ------- is_connected : bool True means the graph is fully connected and False means not. """ if sparse.isspmatrix(graph): # sparse graph, find all the connected components n_connected_components, _ = connected_components(graph) return n_connected_components == 1 else: # dense graph, find all connected components start from node 0 return _graph_connected_component(graph, 0).sum() == graph.shape[0] def _set_diag(laplacian, value, norm_laplacian): """Set the diagonal of the laplacian matrix and convert it to a sparse format well suited for eigenvalue decomposition. Parameters ---------- laplacian : {ndarray, sparse matrix} The graph laplacian. value : float The value of the diagonal. norm_laplacian : bool Whether the value of the diagonal should be changed or not. Returns ------- laplacian : {array, sparse matrix} An array of matrix in a form that is well suited to fast eigenvalue decomposition, depending on the band width of the matrix. """ n_nodes = laplacian.shape[0] # We need all entries in the diagonal to values if not sparse.isspmatrix(laplacian): if norm_laplacian: laplacian.flat[::n_nodes + 1] = value else: laplacian = laplacian.tocoo() if norm_laplacian: diag_idx = (laplacian.row == laplacian.col) laplacian.data[diag_idx] = value # If the matrix has a small number of diagonals (as in the # case of structured matrices coming from images), the # dia format might be best suited for matvec products: n_diags = np.unique(laplacian.row - laplacian.col).size if n_diags <= 7: # 3 or less outer diagonals on each side laplacian = laplacian.todia() else: # csr has the fastest matvec and is thus best suited to # arpack laplacian = laplacian.tocsr() return laplacian @_deprecate_positional_args def 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 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 :ref:`User Guide `. Parameters ---------- adjacency : {array-like, sparse graph} of shape (n_samples, n_samples) The adjacency matrix of the graph to embed. n_components : int, 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_state : int, RandomState instance or None, default=None Determines the random number generator used for the initialization of the lobpcg eigenvectors decomposition when ``solver`` == 'amg'. Pass an int for reproducible results across multiple function calls. See :term: `Glossary `. eigen_tol : float, default=0.0 Stopping criterion for eigendecomposition of the Laplacian matrix when using arpack eigen_solver. norm_laplacian : bool, default=True If True, then compute normalized Laplacian. drop_first : bool, 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 ------- embedding : ndarray 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 https://doi.org/10.1137%2FS1064827500366124 """ adjacency = check_symmetric(adjacency) try: from pyamg import smoothed_aggregation_solver except ImportError as e: if eigen_solver == "amg": raise ValueError("The eigen_solver was set to 'amg', but pyamg is " "not available.") from e if eigen_solver is None: eigen_solver = 'arpack' elif eigen_solver not in ('arpack', 'lobpcg', 'amg'): raise ValueError("Unknown value for eigen_solver: '%s'." "Should be 'amg', 'arpack', or 'lobpcg'" % eigen_solver) random_state = check_random_state(random_state) n_nodes = adjacency.shape[0] # Whether to drop the first eigenvector if drop_first: n_components = n_components + 1 if not _graph_is_connected(adjacency): warnings.warn("Graph is not fully connected, spectral embedding" " may not work as expected.") laplacian, dd = csgraph_laplacian(adjacency, normed=norm_laplacian, return_diag=True) if (eigen_solver == 'arpack' or eigen_solver != 'lobpcg' and (not sparse.isspmatrix(laplacian) or n_nodes < 5 * n_components)): # lobpcg used with eigen_solver='amg' has bugs for low number of nodes # for details see the source code in scipy: # https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen # /lobpcg/lobpcg.py#L237 # or matlab: # https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m laplacian = _set_diag(laplacian, 1, norm_laplacian) # Here we'll use shift-invert mode for fast eigenvalues # (see https://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html # for a short explanation of what this means) # Because the normalized Laplacian has eigenvalues between 0 and 2, # I - L has eigenvalues between -1 and 1. ARPACK is most efficient # when finding eigenvalues of largest magnitude (keyword which='LM') # and when these eigenvalues are very large compared to the rest. # For very large, very sparse graphs, I - L can have many, many # eigenvalues very near 1.0. This leads to slow convergence. So # instead, we'll use ARPACK's shift-invert mode, asking for the # eigenvalues near 1.0. This effectively spreads-out the spectrum # near 1.0 and leads to much faster convergence: potentially an # orders-of-magnitude speedup over simply using keyword which='LA' # in standard mode. try: # We are computing the opposite of the laplacian inplace so as # to spare a memory allocation of a possibly very large array laplacian *= -1 v0 = _init_arpack_v0(laplacian.shape[0], random_state) _, diffusion_map = eigsh( laplacian, k=n_components, sigma=1.0, which='LM', tol=eigen_tol, v0=v0) embedding = diffusion_map.T[n_components::-1] if norm_laplacian: embedding = embedding / dd except RuntimeError: # When submatrices are exactly singular, an LU decomposition # in arpack fails. We fallback to lobpcg eigen_solver = "lobpcg" # Revert the laplacian to its opposite to have lobpcg work laplacian *= -1 elif eigen_solver == 'amg': # Use AMG to get a preconditioner and speed up the eigenvalue # problem. if not sparse.issparse(laplacian): warnings.warn("AMG works better for sparse matrices") # lobpcg needs double precision floats laplacian = check_array(laplacian, dtype=np.float64, accept_sparse=True) laplacian = _set_diag(laplacian, 1, norm_laplacian) # The Laplacian matrix is always singular, having at least one zero # eigenvalue, corresponding to the trivial eigenvector, which is a # constant. Using a singular matrix for preconditioning may result in # random failures in LOBPCG and is not supported by the existing # theory: # see https://doi.org/10.1007/s10208-015-9297-1 # Shift the Laplacian so its diagononal is not all ones. The shift # does change the eigenpairs however, so we'll feed the shifted # matrix to the solver and afterward set it back to the original. diag_shift = 1e-5 * sparse.eye(laplacian.shape[0]) laplacian += diag_shift ml = smoothed_aggregation_solver(check_array(laplacian, accept_sparse='csr')) laplacian -= diag_shift M = ml.aspreconditioner() X = random_state.rand(laplacian.shape[0], n_components + 1) X[:, 0] = dd.ravel() _, diffusion_map = lobpcg(laplacian, X, M=M, tol=1.e-5, largest=False) embedding = diffusion_map.T if norm_laplacian: embedding = embedding / dd if embedding.shape[0] == 1: raise ValueError if eigen_solver == "lobpcg": # lobpcg needs double precision floats laplacian = check_array(laplacian, dtype=np.float64, accept_sparse=True) if n_nodes < 5 * n_components + 1: # see note above under arpack why lobpcg has problems with small # number of nodes # lobpcg will fallback to eigh, so we short circuit it if sparse.isspmatrix(laplacian): laplacian = laplacian.toarray() _, diffusion_map = eigh(laplacian) embedding = diffusion_map.T[:n_components] if norm_laplacian: embedding = embedding / dd else: laplacian = _set_diag(laplacian, 1, norm_laplacian) # We increase the number of eigenvectors requested, as lobpcg # doesn't behave well in low dimension X = random_state.rand(laplacian.shape[0], n_components + 1) X[:, 0] = dd.ravel() _, diffusion_map = lobpcg(laplacian, X, tol=1e-15, largest=False, maxiter=2000) embedding = diffusion_map.T[:n_components] if norm_laplacian: embedding = embedding / dd if embedding.shape[0] == 1: raise ValueError embedding = _deterministic_vector_sign_flip(embedding) if drop_first: return embedding[1:n_components].T else: return embedding[:n_components].T class SpectralEmbedding(BaseEstimator): """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 `. 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 Determines the random number generator used for the initialization of the lobpcg eigenvectors when ``solver`` == 'amg'. Pass an int for reproducible results across multiple function calls. See :term: `Glossary `. 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. 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 :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`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_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 """ @_deprecate_positional_args def __init__(self, n_components=2, *, affinity="nearest_neighbors", gamma=None, random_state=None, eigen_solver=None, n_neighbors=None, n_jobs=None): self.n_components = n_components self.affinity = affinity self.gamma = gamma self.random_state = random_state self.eigen_solver = eigen_solver self.n_neighbors = n_neighbors self.n_jobs = n_jobs def _more_tags(self): return {'pairwise': self.affinity in ["precomputed", "precomputed_nearest_neighbors"]} # TODO: Remove in 1.1 # mypy error: Decorated property not supported @deprecated("Attribute _pairwise was deprecated in " # type: ignore "version 0.24 and will be removed in 1.1 (renaming of 0.26).") @property def _pairwise(self): return self.affinity in ["precomputed", "precomputed_nearest_neighbors"] def _get_affinity_matrix(self, X, Y=None): """Calculate the affinity matrix from data Parameters ---------- X : array-like 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 of shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples. Y: Ignored Returns ------- affinity_matrix of shape (n_samples, n_samples) """ if self.affinity == 'precomputed': self.affinity_matrix_ = X return self.affinity_matrix_ if self.affinity == 'precomputed_nearest_neighbors': estimator = NearestNeighbors(n_neighbors=self.n_neighbors, n_jobs=self.n_jobs, metric="precomputed").fit(X) connectivity = estimator.kneighbors_graph(X=X, mode='connectivity') self.affinity_matrix_ = 0.5 * (connectivity + connectivity.T) return self.affinity_matrix_ if self.affinity == 'nearest_neighbors': if sparse.issparse(X): warnings.warn("Nearest neighbors affinity currently does " "not support sparse input, falling back to " "rbf affinity") self.affinity = "rbf" else: self.n_neighbors_ = (self.n_neighbors if self.n_neighbors is not None else max(int(X.shape[0] / 10), 1)) self.affinity_matrix_ = kneighbors_graph(X, self.n_neighbors_, include_self=True, n_jobs=self.n_jobs) # currently only symmetric affinity_matrix supported self.affinity_matrix_ = 0.5 * (self.affinity_matrix_ + self.affinity_matrix_.T) return self.affinity_matrix_ if self.affinity == 'rbf': self.gamma_ = (self.gamma if self.gamma is not None else 1.0 / X.shape[1]) self.affinity_matrix_ = rbf_kernel(X, gamma=self.gamma_) return self.affinity_matrix_ self.affinity_matrix_ = self.affinity(X) return self.affinity_matrix_ def fit(self, 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 Returns ------- self : object Returns the instance itself. """ X = self._validate_data(X, accept_sparse='csr', ensure_min_samples=2, estimator=self) random_state = check_random_state(self.random_state) if isinstance(self.affinity, str): if self.affinity not in {"nearest_neighbors", "rbf", "precomputed", "precomputed_nearest_neighbors"}: raise ValueError(("%s is not a valid affinity. Expected " "'precomputed', 'rbf', 'nearest_neighbors' " "or a callable.") % self.affinity) elif not callable(self.affinity): raise ValueError(("'affinity' is expected to be an affinity " "name or a callable. Got: %s") % self.affinity) affinity_matrix = self._get_affinity_matrix(X) self.embedding_ = spectral_embedding(affinity_matrix, n_components=self.n_components, eigen_solver=self.eigen_solver, random_state=random_state) return self def fit_transform(self, 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 Returns ------- X_new : array-like of shape (n_samples, n_components) """ self.fit(X) return self.embedding_