import warnings from warnings import warn import numpy as np import scipy.sparse import scipy.sparse.csgraph from sklearn.decomposition import TruncatedSVD from sklearn.manifold import SpectralEmbedding from sklearn.metrics import pairwise_distances from sklearn.metrics.pairwise import _VALID_METRICS as SKLEARN_PAIRWISE_VALID_METRICS from umap.distances import pairwise_special_metric, SPECIAL_METRICS from umap.sparse import SPARSE_SPECIAL_METRICS, sparse_named_distances def component_layout( data, n_components, component_labels, dim, random_state, metric="euclidean", metric_kwds={}, ): """Provide a layout relating the separate connected components. This is done by taking the centroid of each component and then performing a spectral embedding of the centroids. Parameters ---------- data: array of shape (n_samples, n_features) The source data -- required so we can generate centroids for each connected component of the graph. n_components: int The number of distinct components to be layed out. component_labels: array of shape (n_samples) For each vertex in the graph the label of the component to which the vertex belongs. dim: int The chosen embedding dimension. metric: string or callable (optional, default 'euclidean') The metric used to measure distances among the source data points. metric_kwds: dict (optional, default {}) Keyword arguments to be passed to the metric function. If metric is 'precomputed', 'linkage' keyword can be used to specify 'average', 'complete', or 'single' linkage. Default is 'average' Returns ------- component_embedding: array of shape (n_components, dim) The ``dim``-dimensional embedding of the ``n_components``-many connected components. """ if data is None: # We don't have data to work with; just guess return np.random.random(size=(n_components, dim)) * 10.0 component_centroids = np.empty((n_components, data.shape[1]), dtype=np.float64) if metric == "precomputed": # cannot compute centroids from precomputed distances # instead, compute centroid distances using linkage distance_matrix = np.zeros((n_components, n_components), dtype=np.float64) linkage = metric_kwds.get("linkage", "average") if linkage == "average": linkage = np.mean elif linkage == "complete": linkage = np.max elif linkage == "single": linkage = np.min else: raise ValueError( "Unrecognized linkage '%s'. Please choose from " "'average', 'complete', or 'single'" % linkage ) for c_i in range(n_components): dm_i = data[component_labels == c_i] for c_j in range(c_i + 1, n_components): dist = linkage(dm_i[:, component_labels == c_j]) distance_matrix[c_i, c_j] = dist distance_matrix[c_j, c_i] = dist else: for label in range(n_components): component_centroids[label] = data[component_labels == label].mean(axis=0) if scipy.sparse.isspmatrix(component_centroids): warn( "Forcing component centroids to dense; if you are running out of " "memory then consider increasing n_neighbors." ) component_centroids = component_centroids.toarray() if metric in SPECIAL_METRICS: distance_matrix = pairwise_special_metric( component_centroids, metric=metric, kwds=metric_kwds, ) elif metric in SPARSE_SPECIAL_METRICS: distance_matrix = pairwise_special_metric( component_centroids, metric=SPARSE_SPECIAL_METRICS[metric], kwds=metric_kwds, ) else: if callable(metric) and scipy.sparse.isspmatrix(data): function_to_name_mapping = { sparse_named_distances[k]: k for k in set(SKLEARN_PAIRWISE_VALID_METRICS) & set(sparse_named_distances.keys()) } try: metric_name = function_to_name_mapping[metric] except KeyError: raise NotImplementedError( "Multicomponent layout for custom " "sparse metrics is not implemented at " "this time." ) distance_matrix = pairwise_distances( component_centroids, metric=metric_name, **metric_kwds ) else: distance_matrix = pairwise_distances( component_centroids, metric=metric, **metric_kwds ) affinity_matrix = np.exp(-(distance_matrix**2)) component_embedding = SpectralEmbedding( n_components=dim, affinity="precomputed", random_state=random_state ).fit_transform(affinity_matrix) component_embedding /= component_embedding.max() return component_embedding def multi_component_layout( data, graph, n_components, component_labels, dim, random_state, metric="euclidean", metric_kwds={}, init="random", tol=0.0, maxiter=0 ): """Specialised layout algorithm for dealing with graphs with many connected components. This will first find relative positions for the components by spectrally embedding their centroids, then spectrally embed each individual connected component positioning them according to the centroid embeddings. This provides a decent embedding of each component while placing the components in good relative positions to one another. Parameters ---------- data: array of shape (n_samples, n_features) The source data -- required so we can generate centroids for each connected component of the graph. graph: sparse matrix The adjacency matrix of the graph to be embedded. n_components: int The number of distinct components to be layed out. component_labels: array of shape (n_samples) For each vertex in the graph the label of the component to which the vertex belongs. dim: int The chosen embedding dimension. metric: string or callable (optional, default 'euclidean') The metric used to measure distances among the source data points. metric_kwds: dict (optional, default {}) Keyword arguments to be passed to the metric function. init: string, either "random" or "tsvd" Indicates to initialize the eigensolver. Use "random" (the default) to use uniformly distributed random initialization; use "tsvd" to warm-start the eigensolver with singular vectors of the Laplacian associated to the largest singular values. This latter option also forces usage of the LOBPCG eigensolver; with the former, ARPACK's solver ``eigsh`` will be used for smaller Laplacians. tol: float, default chosen by implementation Stopping tolerance for the numerical algorithm computing the embedding. maxiter: int, default chosen by implementation Number of iterations the numerical algorithm will go through at most as it attempts to compute the embedding. Returns ------- embedding: array of shape (n_samples, dim) The initial embedding of ``graph``. """ result = np.empty((graph.shape[0], dim), dtype=np.float32) if n_components > 2 * dim: meta_embedding = component_layout( data, n_components, component_labels, dim, random_state, metric=metric, metric_kwds=metric_kwds, ) else: k = int(np.ceil(n_components / 2.0)) base = np.hstack([np.eye(k), np.zeros((k, dim - k))]) meta_embedding = np.vstack([base, -base])[:n_components] for label in range(n_components): component_graph = graph.tocsr()[component_labels == label, :].tocsc() component_graph = component_graph[:, component_labels == label].tocoo() distances = pairwise_distances([meta_embedding[label]], meta_embedding) data_range = distances[distances > 0.0].min() / 2.0 if component_graph.shape[0] < 2 * dim or component_graph.shape[0] <= dim + 1: result[component_labels == label] = ( random_state.uniform( low=-data_range, high=data_range, size=(component_graph.shape[0], dim), ) + meta_embedding[label] ) else: component_embedding = _spectral_layout( data=None, graph=component_graph, dim=dim, random_state=random_state, metric=metric, metric_kwds=metric_kwds, init=init, tol=tol, maxiter=maxiter ) expansion = data_range / np.max(np.abs(component_embedding)) component_embedding *= expansion result[component_labels == label] = ( component_embedding + meta_embedding[label] ) return result def spectral_layout( data, graph, dim, random_state, metric="euclidean", metric_kwds={}, tol=0.0, maxiter=0 ): """ Given a graph compute the spectral embedding of the graph. This is simply the eigenvectors of the laplacian of the graph. Here we use the normalized laplacian. Parameters ---------- data: array of shape (n_samples, n_features) The source data graph: sparse matrix The (weighted) adjacency matrix of the graph as a sparse matrix. dim: int The dimension of the space into which to embed. random_state: numpy RandomState or equivalent A state capable being used as a numpy random state. tol: float, default chosen by implementation Stopping tolerance for the numerical algorithm computing the embedding. maxiter: int, default chosen by implementation Number of iterations the numerical algorithm will go through at most as it attempts to compute the embedding. Returns ------- embedding: array of shape (n_vertices, dim) The spectral embedding of the graph. """ return _spectral_layout( data=data, graph=graph, dim=dim, random_state=random_state, metric=metric, metric_kwds=metric_kwds, init="random", tol=tol, maxiter=maxiter ) def tswspectral_layout( data, graph, dim, random_state, metric="euclidean", metric_kwds={}, method=None, tol=0.0, maxiter=0 ): """Given a graph, compute the spectral embedding of the graph. This is simply the eigenvectors of the Laplacian of the graph. Here we use the normalized laplacian and a truncated SVD-based guess of the eigenvectors to "warm" up the eigensolver. This function should give results of similar accuracy to the spectral_layout function, but may converge more quickly for graph Laplacians that cause spectral_layout to take an excessive amount of time to complete. Parameters ---------- data: array of shape (n_samples, n_features) The source data graph: sparse matrix The (weighted) adjacency matrix of the graph as a sparse matrix. dim: int The dimension of the space into which to embed. random_state: numpy RandomState or equivalent A state capable being used as a numpy random state. metric: string or callable (optional, default 'euclidean') The metric used to measure distances among the source data points. Used only if the multiple connected components are found in the graph. metric_kwds: dict (optional, default {}) Keyword arguments to be passed to the metric function. If metric is 'precomputed', 'linkage' keyword can be used to specify 'average', 'complete', or 'single' linkage. Default is 'average'. Used only if the multiple connected components are found in the graph. method: str (optional, default None, values either 'eigsh' or 'lobpcg') Name of the eigenvalue computation method to use to compute the spectral embedding. If left to None (or empty string), as by default, the method is chosen from the number of vectors in play: larger vector collections are handled with lobpcg, smaller collections with eigsh. Method names correspond to SciPy routines in scipy.sparse.linalg. tol: float, default chosen by implementation Stopping tolerance for the numerical algorithm computing the embedding. maxiter: int, default chosen by implementation Number of iterations the numerical algorithm will go through at most as it attempts to compute the embedding. Returns ------- embedding: array of shape (n_vertices, dim) The spectral embedding of the graph. """ return _spectral_layout( data=data, graph=graph, dim=dim, random_state=random_state, metric=metric, metric_kwds=metric_kwds, init="tsvd", method=method, tol=tol, maxiter=maxiter ) def _spectral_layout( data, graph, dim, random_state, metric="euclidean", metric_kwds={}, init="random", method=None, tol=0.0, maxiter=0 ): """General implementation of the spectral embedding of the graph, derived as a subset of the eigenvectors of the normalized Laplacian of the graph. The numerical method for computing the eigendecomposition is chosen through heuristics. Parameters ---------- data: array of shape (n_samples, n_features) The source data graph: sparse matrix The (weighted) adjacency matrix of the graph as a sparse matrix. dim: int The dimension of the space into which to embed. random_state: numpy RandomState or equivalent A state capable being used as a numpy random state. metric: string or callable (optional, default 'euclidean') The metric used to measure distances among the source data points. Used only if the multiple connected components are found in the graph. metric_kwds: dict (optional, default {}) Keyword arguments to be passed to the metric function. If metric is 'precomputed', 'linkage' keyword can be used to specify 'average', 'complete', or 'single' linkage. Default is 'average'. Used only if the multiple connected components are found in the graph. init: string, either "random" or "tsvd" Indicates to initialize the eigensolver. Use "random" (the default) to use uniformly distributed random initialization; use "tsvd" to warm-start the eigensolver with singular vectors of the Laplacian associated to the largest singular values. This latter option also forces usage of the LOBPCG eigensolver; with the former, ARPACK's solver ``eigsh`` will be used for smaller Laplacians. method: string -- either "eigsh" or "lobpcg" -- or None Name of the eigenvalue computation method to use to compute the spectral embedding. If left to None (or empty string), as by default, the method is chosen from the number of vectors in play: larger vector collections are handled with lobpcg, smaller collections with eigsh. Method names correspond to SciPy routines in scipy.sparse.linalg. tol: float, default chosen by implementation Stopping tolerance for the numerical algorithm computing the embedding. maxiter: int, default chosen by implementation Number of iterations the numerical algorithm will go through at most as it attempts to compute the embedding. Returns ------- embedding: array of shape (n_vertices, dim) The spectral embedding of the graph. """ n_samples = graph.shape[0] n_components, labels = scipy.sparse.csgraph.connected_components(graph) if n_components > 1: return multi_component_layout( data, graph, n_components, labels, dim, random_state, metric=metric, metric_kwds=metric_kwds, ) sqrt_deg = np.sqrt(np.asarray(graph.sum(axis=0)).squeeze()) # standard Laplacian # D = scipy.sparse.spdiags(diag_data, 0, graph.shape[0], graph.shape[0]) # L = D - graph # Normalized Laplacian I = scipy.sparse.identity(graph.shape[0], dtype=np.float64) D = scipy.sparse.spdiags( 1.0 / sqrt_deg, 0, graph.shape[0], graph.shape[0] ) L = I - D * graph * D if not scipy.sparse.issparse(L): L = np.asarray(L) k = dim + 1 num_lanczos_vectors = max(2 * k + 1, int(np.sqrt(graph.shape[0]))) gen = ( random_state if isinstance(random_state, (np.random.Generator, np.random.RandomState)) else np.random.default_rng(seed=random_state) ) if not method: method = "eigsh" if L.shape[0] < 2000000 else "lobpcg" try: if init == "random": X = gen.normal(size=(L.shape[0], k)) elif init == "tsvd": X = TruncatedSVD( n_components=k, random_state=random_state, # algorithm="arpack" ).fit_transform(L) else: raise ValueError( "The init parameter must be either 'random' or 'tsvd': " f"{init} is invalid." ) # For such a normalized Laplacian, the first eigenvector is always # proportional to sqrt(degrees). We thus replace the first t-SVD guess # with the exact value. X[:, 0] = sqrt_deg / np.linalg.norm(sqrt_deg) if method == "eigsh": eigenvalues, eigenvectors = scipy.sparse.linalg.eigsh( L, k, which="SM", ncv=num_lanczos_vectors, tol=tol or 1e-4, v0=np.ones(L.shape[0]), maxiter=maxiter or graph.shape[0] * 5, ) elif method == "lobpcg": with warnings.catch_warnings(): warnings.filterwarnings( category=UserWarning, message=r"(?ms).*not reaching the requested tolerance", action="error" ) eigenvalues, eigenvectors = scipy.sparse.linalg.lobpcg( L, np.asarray(X), largest=False, tol=tol or 1e-4, maxiter=maxiter or 5 * graph.shape[0] ) else: raise ValueError("Method should either be None, 'eigsh' or 'lobpcg'") order = np.argsort(eigenvalues)[1:k] return eigenvectors[:, order] except (scipy.sparse.linalg.ArpackError, UserWarning): warn( "Spectral initialisation failed! The eigenvector solver\n" "failed. This is likely due to too small an eigengap. Consider\n" "adding some noise or jitter to your data.\n\n" "Falling back to random initialisation!" ) return gen.uniform(low=-10.0, high=10.0, size=(graph.shape[0], dim))