"""Isomap for manifold learning""" # Author: Jake Vanderplas -- # License: BSD 3 clause (C) 2011 import numpy as np from ..base import BaseEstimator, TransformerMixin from ..neighbors import NearestNeighbors, kneighbors_graph from ..utils.validation import check_is_fitted from ..utils.validation import _deprecate_positional_args from ..utils.graph import graph_shortest_path from ..decomposition import KernelPCA from ..preprocessing import KernelCenterer class Isomap(TransformerMixin, BaseEstimator): """Isomap Embedding Non-linear dimensionality reduction through Isometric Mapping Read more in the :ref:`User Guide `. Parameters ---------- n_neighbors : int, default=5 number of neighbors to consider for each point. n_components : int, default=2 number of coordinates for the manifold eigen_solver : {'auto', 'arpack', 'dense'}, default='auto' 'auto' : Attempt to choose the most efficient solver for the given problem. 'arpack' : Use Arnoldi decomposition to find the eigenvalues and eigenvectors. 'dense' : Use a direct solver (i.e. LAPACK) for the eigenvalue decomposition. tol : float, default=0 Convergence tolerance passed to arpack or lobpcg. not used if eigen_solver == 'dense'. max_iter : int, default=None Maximum number of iterations for the arpack solver. not used if eigen_solver == 'dense'. path_method : {'auto', 'FW', 'D'}, default='auto' Method to use in finding shortest path. 'auto' : attempt to choose the best algorithm automatically. 'FW' : Floyd-Warshall algorithm. 'D' : Dijkstra's algorithm. neighbors_algorithm : {'auto', 'brute', 'kd_tree', 'ball_tree'}, \ default='auto' Algorithm to use for nearest neighbors search, passed to neighbors.NearestNeighbors instance. n_jobs : int or None, 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. metric : string, or callable, default="minkowski" The metric to use when calculating distance between instances in a feature array. If metric is a string or callable, it must be one of the options allowed by :func:`sklearn.metrics.pairwise_distances` for its metric parameter. If metric is "precomputed", X is assumed to be a distance matrix and must be square. X may be a :term:`Glossary `. .. versionadded:: 0.22 p : int, default=2 Parameter for the Minkowski metric from sklearn.metrics.pairwise.pairwise_distances. When p = 1, this is equivalent to using manhattan_distance (l1), and euclidean_distance (l2) for p = 2. For arbitrary p, minkowski_distance (l_p) is used. .. versionadded:: 0.22 metric_params : dict, default=None Additional keyword arguments for the metric function. .. versionadded:: 0.22 Attributes ---------- embedding_ : array-like, shape (n_samples, n_components) Stores the embedding vectors. kernel_pca_ : object :class:`~sklearn.decomposition.KernelPCA` object used to implement the embedding. nbrs_ : sklearn.neighbors.NearestNeighbors instance Stores nearest neighbors instance, including BallTree or KDtree if applicable. dist_matrix_ : array-like, shape (n_samples, n_samples) Stores the geodesic distance matrix of training data. Examples -------- >>> from sklearn.datasets import load_digits >>> from sklearn.manifold import Isomap >>> X, _ = load_digits(return_X_y=True) >>> X.shape (1797, 64) >>> embedding = Isomap(n_components=2) >>> X_transformed = embedding.fit_transform(X[:100]) >>> X_transformed.shape (100, 2) References ---------- .. [1] Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. A global geometric framework for nonlinear dimensionality reduction. Science 290 (5500) """ @_deprecate_positional_args def __init__(self, *, n_neighbors=5, n_components=2, eigen_solver='auto', tol=0, max_iter=None, path_method='auto', neighbors_algorithm='auto', n_jobs=None, metric='minkowski', p=2, metric_params=None): self.n_neighbors = n_neighbors self.n_components = n_components self.eigen_solver = eigen_solver self.tol = tol self.max_iter = max_iter self.path_method = path_method self.neighbors_algorithm = neighbors_algorithm self.n_jobs = n_jobs self.metric = metric self.p = p self.metric_params = metric_params def _fit_transform(self, X): self.nbrs_ = NearestNeighbors(n_neighbors=self.n_neighbors, algorithm=self.neighbors_algorithm, metric=self.metric, p=self.p, metric_params=self.metric_params, n_jobs=self.n_jobs) self.nbrs_.fit(X) self.n_features_in_ = self.nbrs_.n_features_in_ self.kernel_pca_ = KernelPCA(n_components=self.n_components, kernel="precomputed", eigen_solver=self.eigen_solver, tol=self.tol, max_iter=self.max_iter, n_jobs=self.n_jobs) kng = kneighbors_graph(self.nbrs_, self.n_neighbors, metric=self.metric, p=self.p, metric_params=self.metric_params, mode='distance', n_jobs=self.n_jobs) self.dist_matrix_ = graph_shortest_path(kng, method=self.path_method, directed=False) G = self.dist_matrix_ ** 2 G *= -0.5 self.embedding_ = self.kernel_pca_.fit_transform(G) def reconstruction_error(self): """Compute the reconstruction error for the embedding. Returns ------- reconstruction_error : float Notes ----- The cost function of an isomap embedding is ``E = frobenius_norm[K(D) - K(D_fit)] / n_samples`` Where D is the matrix of distances for the input data X, D_fit is the matrix of distances for the output embedding X_fit, and K is the isomap kernel: ``K(D) = -0.5 * (I - 1/n_samples) * D^2 * (I - 1/n_samples)`` """ G = -0.5 * self.dist_matrix_ ** 2 G_center = KernelCenterer().fit_transform(G) evals = self.kernel_pca_.lambdas_ return np.sqrt(np.sum(G_center ** 2) - np.sum(evals ** 2)) / G.shape[0] def fit(self, X, y=None): """Compute the embedding vectors for data X Parameters ---------- X : {array-like, sparse graph, BallTree, KDTree, NearestNeighbors} Sample data, shape = (n_samples, n_features), in the form of a numpy array, sparse graph, precomputed tree, or NearestNeighbors object. y : Ignored Returns ------- self : returns an instance of self. """ self._fit_transform(X) return self def fit_transform(self, X, y=None): """Fit the model from data in X and transform X. Parameters ---------- X : {array-like, sparse graph, BallTree, KDTree} Training vector, where n_samples in the number of samples and n_features is the number of features. y : Ignored Returns ------- X_new : array-like, shape (n_samples, n_components) """ self._fit_transform(X) return self.embedding_ def transform(self, X): """Transform X. This is implemented by linking the points X into the graph of geodesic distances of the training data. First the `n_neighbors` nearest neighbors of X are found in the training data, and from these the shortest geodesic distances from each point in X to each point in the training data are computed in order to construct the kernel. The embedding of X is the projection of this kernel onto the embedding vectors of the training set. Parameters ---------- X : array-like, shape (n_queries, n_features) If neighbors_algorithm='precomputed', X is assumed to be a distance matrix or a sparse graph of shape (n_queries, n_samples_fit). Returns ------- X_new : array-like, shape (n_queries, n_components) """ check_is_fitted(self) distances, indices = self.nbrs_.kneighbors(X, return_distance=True) # Create the graph of shortest distances from X to # training data via the nearest neighbors of X. # This can be done as a single array operation, but it potentially # takes a lot of memory. To avoid that, use a loop: n_samples_fit = self.nbrs_.n_samples_fit_ n_queries = distances.shape[0] G_X = np.zeros((n_queries, n_samples_fit)) for i in range(n_queries): G_X[i] = np.min(self.dist_matrix_[indices[i]] + distances[i][:, None], 0) G_X **= 2 G_X *= -0.5 return self.kernel_pca_.transform(G_X)