439 lines
15 KiB
Python
439 lines
15 KiB
Python
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"""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 warnings
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from numbers import Integral, Real
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import numpy as np
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from scipy.sparse import issparse
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from scipy.sparse.csgraph import connected_components, shortest_path
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from ..base import (
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BaseEstimator,
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ClassNamePrefixFeaturesOutMixin,
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TransformerMixin,
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_fit_context,
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)
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from ..decomposition import KernelPCA
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from ..metrics.pairwise import _VALID_METRICS
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from ..neighbors import NearestNeighbors, kneighbors_graph, radius_neighbors_graph
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from ..preprocessing import KernelCenterer
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from ..utils._param_validation import Interval, StrOptions
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from ..utils.graph import _fix_connected_components
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from ..utils.validation import check_is_fitted
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class Isomap(ClassNamePrefixFeaturesOutMixin, 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 or None, default=5
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Number of neighbors to consider for each point. If `n_neighbors` is an int,
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then `radius` must be `None`.
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radius : float or None, default=None
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Limiting distance of neighbors to return. If `radius` is a float,
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then `n_neighbors` must be set to `None`.
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.. versionadded:: 1.1
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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 : str, 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 : float, 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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n_features_in_ : int
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Number of features seen during :term:`fit`.
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.. versionadded:: 0.24
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feature_names_in_ : ndarray of shape (`n_features_in_`,)
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Names of features seen during :term:`fit`. Defined only when `X`
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has feature names that are all strings.
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.. versionadded:: 1.0
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See Also
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--------
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sklearn.decomposition.PCA : Principal component analysis that is a linear
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dimensionality reduction method.
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sklearn.decomposition.KernelPCA : Non-linear dimensionality reduction using
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kernels and PCA.
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MDS : Manifold learning using multidimensional scaling.
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TSNE : T-distributed Stochastic Neighbor Embedding.
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LocallyLinearEmbedding : Manifold learning using Locally Linear Embedding.
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SpectralEmbedding : Spectral embedding for non-linear dimensionality.
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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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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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"""
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_parameter_constraints: dict = {
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"n_neighbors": [Interval(Integral, 1, None, closed="left"), None],
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"radius": [Interval(Real, 0, None, closed="both"), None],
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"n_components": [Interval(Integral, 1, None, closed="left")],
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"eigen_solver": [StrOptions({"auto", "arpack", "dense"})],
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"tol": [Interval(Real, 0, None, closed="left")],
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"max_iter": [Interval(Integral, 1, None, closed="left"), None],
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"path_method": [StrOptions({"auto", "FW", "D"})],
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"neighbors_algorithm": [StrOptions({"auto", "brute", "kd_tree", "ball_tree"})],
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"n_jobs": [Integral, None],
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"p": [Interval(Real, 1, None, closed="left")],
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"metric": [StrOptions(set(_VALID_METRICS) | {"precomputed"}), callable],
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"metric_params": [dict, None],
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}
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def __init__(
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self,
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*,
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n_neighbors=5,
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radius=None,
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n_components=2,
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eigen_solver="auto",
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tol=0,
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max_iter=None,
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path_method="auto",
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neighbors_algorithm="auto",
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n_jobs=None,
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metric="minkowski",
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p=2,
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metric_params=None,
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):
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self.n_neighbors = n_neighbors
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self.radius = radius
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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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if self.n_neighbors is not None and self.radius is not None:
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raise ValueError(
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"Both n_neighbors and radius are provided. Use"
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f" Isomap(radius={self.radius}, n_neighbors=None) if intended to use"
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" radius-based neighbors"
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)
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self.nbrs_ = NearestNeighbors(
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n_neighbors=self.n_neighbors,
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radius=self.radius,
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algorithm=self.neighbors_algorithm,
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metric=self.metric,
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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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)
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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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if hasattr(self.nbrs_, "feature_names_in_"):
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self.feature_names_in_ = self.nbrs_.feature_names_in_
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self.kernel_pca_ = KernelPCA(
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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,
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max_iter=self.max_iter,
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n_jobs=self.n_jobs,
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).set_output(transform="default")
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if self.n_neighbors is not None:
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nbg = kneighbors_graph(
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self.nbrs_,
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self.n_neighbors,
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metric=self.metric,
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p=self.p,
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metric_params=self.metric_params,
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mode="distance",
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n_jobs=self.n_jobs,
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)
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else:
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nbg = radius_neighbors_graph(
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self.nbrs_,
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radius=self.radius,
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metric=self.metric,
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p=self.p,
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metric_params=self.metric_params,
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mode="distance",
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n_jobs=self.n_jobs,
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)
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# Compute the number of connected components, and connect the different
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# components to be able to compute a shortest path between all pairs
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# of samples in the graph.
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# Similar fix to cluster._agglomerative._fix_connectivity.
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n_connected_components, labels = connected_components(nbg)
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if n_connected_components > 1:
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if self.metric == "precomputed" and issparse(X):
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raise RuntimeError(
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"The number of connected components of the neighbors graph"
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f" is {n_connected_components} > 1. The graph cannot be "
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"completed with metric='precomputed', and Isomap cannot be"
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"fitted. Increase the number of neighbors to avoid this "
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"issue, or precompute the full distance matrix instead "
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"of passing a sparse neighbors graph."
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)
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warnings.warn(
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(
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"The number of connected components of the neighbors graph "
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f"is {n_connected_components} > 1. Completing the graph to fit"
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" Isomap might be slow. Increase the number of neighbors to "
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"avoid this issue."
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),
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stacklevel=2,
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)
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# use array validated by NearestNeighbors
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nbg = _fix_connected_components(
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X=self.nbrs_._fit_X,
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graph=nbg,
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n_connected_components=n_connected_components,
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component_labels=labels,
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mode="distance",
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metric=self.nbrs_.effective_metric_,
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**self.nbrs_.effective_metric_params_,
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)
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self.dist_matrix_ = shortest_path(nbg, method=self.path_method, directed=False)
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if self.nbrs_._fit_X.dtype == np.float32:
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self.dist_matrix_ = self.dist_matrix_.astype(
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self.nbrs_._fit_X.dtype, copy=False
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)
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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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self._n_features_out = self.embedding_.shape[1]
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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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Reconstruction error.
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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_.eigenvalues_
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return np.sqrt(np.sum(G_center**2) - np.sum(evals**2)) / G.shape[0]
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@_fit_context(
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# Isomap.metric is not validated yet
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prefer_skip_nested_validation=False
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)
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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 matrix, 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 matrix, precomputed tree, or NearestNeighbors
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object.
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y : Ignored
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Not used, present for API consistency by convention.
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Returns
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-------
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self : object
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Returns a fitted 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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@_fit_context(
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# Isomap.metric is not validated yet
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prefer_skip_nested_validation=False
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)
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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 matrix, BallTree, KDTree}
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Training vector, where `n_samples` is the number of samples
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and `n_features` is the number of features.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
Not used, present for API consistency by convention.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
X_new : array-like, shape (n_samples, n_components)
|
||
|
|
X transformed in the new space.
|
||
|
|
"""
|
||
|
|
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, sparse matrix}, 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)
|
||
|
|
X transformed in the new space.
|
||
|
|
"""
|
||
|
|
check_is_fitted(self)
|
||
|
|
if self.n_neighbors is not None:
|
||
|
|
distances, indices = self.nbrs_.kneighbors(X, return_distance=True)
|
||
|
|
else:
|
||
|
|
distances, indices = self.nbrs_.radius_neighbors(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]
|
||
|
|
|
||
|
|
if hasattr(X, "dtype") and X.dtype == np.float32:
|
||
|
|
dtype = np.float32
|
||
|
|
else:
|
||
|
|
dtype = np.float64
|
||
|
|
|
||
|
|
G_X = np.zeros((n_queries, n_samples_fit), dtype)
|
||
|
|
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)
|
||
|
|
|
||
|
|
def _more_tags(self):
|
||
|
|
return {"preserves_dtype": [np.float64, np.float32]}
|