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