"""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 lobpcg from ..base import BaseEstimator from ..externals import six from ..utils import check_random_state, check_array, check_symmetric from ..utils.extmath import _deterministic_vector_sign_flip from ..utils.graph import graph_laplacian from ..utils.sparsetools import connected_components from ..utils.arpack import eigsh from ..metrics.pairwise import rbf_kernel from ..neighbors import kneighbors_graph def _graph_connected_component(graph, node_id): """Find the largest graph connected components that contains one given node Parameters ---------- graph : array-like, 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, 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 """ connected_components_matrix = np.zeros( shape=(graph.shape[0]), dtype=np.bool) connected_components_matrix[node_id] = True n_node = graph.shape[0] for i in range(n_node): last_num_component = connected_components_matrix.sum() _, node_to_add = np.where(graph[connected_components_matrix] != 0) connected_components_matrix[node_to_add] = True if last_num_component >= connected_components_matrix.sum(): break return connected_components_matrix def _graph_is_connected(graph): """ Return whether the graph is connected (True) or Not (False) Parameters ---------- graph : array-like or sparse matrix, 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): """Set the diagonal of the laplacian matrix and convert it to a sparse format well suited for eigenvalue decomposition Parameters ---------- laplacian : array or sparse matrix The graph laplacian value : float The value of the diagonal Returns ------- laplacian : array or 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): laplacian.flat[::n_nodes + 1] = value else: laplacian = laplacian.tocoo() 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 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. Read more in the :ref:`User Guide `. 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 eigenvectors 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 """ adjacency = check_symmetric(adjacency) try: from pyamg import smoothed_aggregation_solver except ImportError: if eigen_solver == "amg": raise ValueError("The eigen_solver was set to 'amg', but pyamg is " "not available.") 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 = graph_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: # http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m laplacian = _set_diag(laplacian, 1) # Here we'll use shift-invert mode for fast eigenvalues # (see http://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 lambdas, diffusion_map = eigsh(laplacian, k=n_components, sigma=1.0, which='LM', tol=eigen_tol) embedding = diffusion_map.T[n_components::-1] * 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 if 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) ml = smoothed_aggregation_solver(check_array(laplacian, 'csr')) M = ml.aspreconditioner() X = random_state.rand(laplacian.shape[0], n_components + 1) X[:, 0] = dd.ravel() lambdas, diffusion_map = lobpcg(laplacian, X, M=M, tol=1.e-12, largest=False) embedding = diffusion_map.T * dd if embedding.shape[0] == 1: raise ValueError elif 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() lambdas, diffusion_map = eigh(laplacian) embedding = diffusion_map.T[:n_components] * dd else: laplacian = _set_diag(laplacian, 1) # 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() lambdas, diffusion_map = lobpcg(laplacian, X, tol=1e-15, largest=False, maxiter=2000) embedding = diffusion_map.T[:n_components] * 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. Read more in the :ref:`User Guide `. Parameters ----------- n_components : integer, default: 2 The dimension of the projected subspace. 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, but may also lead to instabilities. random_state : int seed, RandomState instance, or None, default : None A pseudo random number generator used for the initialization of the lobpcg eigenvectors decomposition when eigen_solver == 'amg'. affinity : string or callable, default : "nearest_neighbors" How to construct the affinity matrix. - 'nearest_neighbors' : construct affinity matrix by knn graph - 'rbf' : construct affinity matrix by rbf kernel - 'precomputed' : interpret X as precomputed affinity matrix - 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. n_neighbors : int, default : max(n_samples/10 , 1) Number of nearest neighbors for nearest_neighbors graph building. 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. 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, 2011 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 """ def __init__(self, n_components=2, affinity="nearest_neighbors", gamma=None, random_state=None, eigen_solver=None, n_neighbors=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 @property def _pairwise(self): return self.affinity == "precomputed" def _get_affinity_matrix(self, X, Y=None): """Calculate the affinity matrix from data Parameters ---------- X : array-like, 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, shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples. Returns ------- affinity_matrix, shape (n_samples, n_samples) """ if self.affinity == 'precomputed': self.affinity_matrix_ = X 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) # 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, 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, shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples. Returns ------- self : object Returns the instance itself. """ X = check_array(X, ensure_min_samples=2, estimator=self) random_state = check_random_state(self.random_state) if isinstance(self.affinity, six.string_types): if self.affinity not in set(("nearest_neighbors", "rbf", "precomputed")): 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 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, 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, shape (n_samples, n_samples), Interpret X as precomputed adjacency graph computed from samples. Returns ------- X_new: array-like, shape (n_samples, n_components) """ self.fit(X) return self.embedding_