"""Spectral Embedding.""" # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause import warnings from numbers import Integral, Real import numpy as np from scipy import sparse from scipy.linalg import eigh from scipy.sparse.csgraph import connected_components from scipy.sparse.linalg import eigsh, lobpcg from ..base import BaseEstimator, _fit_context from ..metrics.pairwise import rbf_kernel from ..neighbors import NearestNeighbors, kneighbors_graph from ..utils import ( check_array, check_random_state, check_symmetric, ) from ..utils._arpack import _init_arpack_v0 from ..utils._param_validation import Interval, StrOptions, validate_params from ..utils.extmath import _deterministic_vector_sign_flip from ..utils.fixes import laplacian as csgraph_laplacian from ..utils.fixes import parse_version, sp_version from ..utils.validation import validate_data 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): # scipy not yet implemented 1D sparse slices; can be changed back to # `neighbors = graph[i].toarray().ravel()` once implemented 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.issparse(graph): # Before Scipy 1.11.3, `connected_components` only supports 32-bit indices. # PR: https://github.com/scipy/scipy/pull/18913 # First integration in 1.11.3: https://github.com/scipy/scipy/pull/19279 # TODO(jjerphan): Once SciPy 1.11.3 is the minimum supported version, use # `accept_large_sparse=True`. accept_large_sparse = sp_version >= parse_version("1.11.3") graph = check_array( graph, accept_sparse=True, accept_large_sparse=accept_large_sparse ) # 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.issparse(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 @validate_params( { "adjacency": ["array-like", "sparse matrix"], "n_components": [Interval(Integral, 1, None, closed="left")], "eigen_solver": [StrOptions({"arpack", "lobpcg", "amg"}), None], "random_state": ["random_state"], "eigen_tol": [Interval(Real, 0, None, closed="left"), StrOptions({"auto"})], "norm_laplacian": ["boolean"], "drop_first": ["boolean"], }, prefer_skip_nested_validation=True, ) def 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 :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 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 :term:`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 https://github.com/pyamg/pyamg/issues/139 for further information. eigen_tol : float, 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, :func:`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. .. versionadded:: 1.2 Added 'auto' option. norm_laplacian : bool, default=True If True, then compute symmetric 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 * :doi:`"Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method", Andrew V. Knyazev <10.1137/S1064827500366124>` 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) """ random_state = check_random_state(random_state) return _spectral_embedding( adjacency, n_components=n_components, eigen_solver=eigen_solver, random_state=random_state, eigen_tol=eigen_tol, norm_laplacian=norm_laplacian, drop_first=drop_first, ) def _spectral_embedding( adjacency, *, n_components=8, eigen_solver=None, random_state=None, eigen_tol="auto", norm_laplacian=True, drop_first=True, ): adjacency = check_symmetric(adjacency) if eigen_solver == "amg": try: from pyamg import smoothed_aggregation_solver except ImportError as e: raise ValueError( "The eigen_solver was set to 'amg', but pyamg is not available." ) from e if eigen_solver is None: eigen_solver = "arpack" 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.issparse(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 tol = 0 if eigen_tol == "auto" else eigen_tol laplacian *= -1 v0 = _init_arpack_v0(laplacian.shape[0], random_state) laplacian = check_array( laplacian, accept_sparse="csr", accept_large_sparse=False ) _, diffusion_map = eigsh( laplacian, k=n_components, sigma=1.0, which="LM", tol=tol, v0=v0 ) embedding = diffusion_map.T[n_components::-1] if norm_laplacian: # recover u = D^-1/2 x from the eigenvector output x 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") laplacian = check_array( laplacian, dtype=[np.float64, np.float32], 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 if hasattr(sparse, "csr_array") and isinstance(laplacian, sparse.csr_array): # `pyamg` does not work with `csr_array` and we need to convert it to a # `csr_matrix` object. laplacian = sparse.csr_matrix(laplacian) ml = smoothed_aggregation_solver(check_array(laplacian, accept_sparse="csr")) laplacian -= diag_shift M = ml.aspreconditioner() # Create initial approximation X to eigenvectors X = random_state.standard_normal(size=(laplacian.shape[0], n_components + 1)) X[:, 0] = dd.ravel() X = X.astype(laplacian.dtype) tol = None if eigen_tol == "auto" else eigen_tol _, diffusion_map = lobpcg(laplacian, X, M=M, tol=tol, largest=False) embedding = diffusion_map.T if norm_laplacian: # recover u = D^-1/2 x from the eigenvector output x embedding = embedding / dd if embedding.shape[0] == 1: raise ValueError if eigen_solver == "lobpcg": laplacian = check_array( laplacian, dtype=[np.float64, np.float32], 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.issparse(laplacian): laplacian = laplacian.toarray() _, diffusion_map = eigh(laplacian, check_finite=False) embedding = diffusion_map.T[:n_components] if norm_laplacian: # recover u = D^-1/2 x from the eigenvector output x 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 and create initial # approximation X to eigenvectors X = random_state.standard_normal( size=(laplacian.shape[0], n_components + 1) ) X[:, 0] = dd.ravel() X = X.astype(laplacian.dtype) tol = None if eigen_tol == "auto" else eigen_tol _, diffusion_map = lobpcg( laplacian, X, tol=tol, largest=False, maxiter=2000 ) embedding = diffusion_map.T[:n_components] if norm_laplacian: # recover u = D^-1/2 x from the eigenvector output x 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 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 :term:`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 https://github.com/pyamg/pyamg/issues/139 for further information. 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. eigen_tol : float, 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, :func:`scipy.sparse.linalg.lobpcg` for details. Note that when using `eigen_solver="lobpcg"` or `eigen_solver="amg"` values of `tol<1e-5` may lead to convergence issues and should be avoided. .. versionadded:: 1.2 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_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 n_neighbors_ : int Number of nearest neighbors effectively used. See Also -------- Isomap : Non-linear dimensionality reduction through Isometric Mapping. References ---------- - :doi:`A Tutorial on Spectral Clustering, 2007 Ulrike von Luxburg <10.1007/s11222-007-9033-z>` - `On Spectral Clustering: Analysis and an algorithm, 2001 Andrew Y. Ng, Michael I. Jordan, Yair Weiss `_ - :doi:`Normalized cuts and image segmentation, 2000 Jianbo Shi, Jitendra Malik <10.1109/34.868688>` 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) """ _parameter_constraints: dict = { "n_components": [Interval(Integral, 1, None, closed="left")], "affinity": [ StrOptions( { "nearest_neighbors", "rbf", "precomputed", "precomputed_nearest_neighbors", }, ), callable, ], "gamma": [Interval(Real, 0, None, closed="left"), None], "random_state": ["random_state"], "eigen_solver": [StrOptions({"arpack", "lobpcg", "amg"}), None], "eigen_tol": [Interval(Real, 0, None, closed="left"), StrOptions({"auto"})], "n_neighbors": [Interval(Integral, 1, None, closed="left"), None], "n_jobs": [None, Integral], } def __init__( self, n_components=2, *, affinity="nearest_neighbors", gamma=None, random_state=None, eigen_solver=None, eigen_tol="auto", 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.eigen_tol = eigen_tol self.n_neighbors = n_neighbors self.n_jobs = n_jobs def __sklearn_tags__(self): tags = super().__sklearn_tags__() tags.input_tags.sparse = True tags.input_tags.pairwise = self.affinity in [ "precomputed", "precomputed_nearest_neighbors", ] return tags 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_ @_fit_context(prefer_skip_nested_validation=True) 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 Not used, present for API consistency by convention. Returns ------- self : object Returns the instance itself. """ X = validate_data(self, X, accept_sparse="csr", ensure_min_samples=2) random_state = check_random_state(self.random_state) affinity_matrix = self._get_affinity_matrix(X) self.embedding_ = _spectral_embedding( affinity_matrix, n_components=self.n_components, eigen_solver=self.eigen_solver, eigen_tol=self.eigen_tol, 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 Not used, present for API consistency by convention. Returns ------- X_new : array-like of shape (n_samples, n_components) Spectral embedding of the training matrix. """ self.fit(X) return self.embedding_