715 lines
27 KiB
Python
715 lines
27 KiB
Python
"""Spectral Embedding."""
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# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
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# Wei LI <kuantkid@gmail.com>
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# License: BSD 3 clause
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from numbers import Integral, Real
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import warnings
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import numpy as np
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from scipy import sparse
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from scipy.linalg import eigh
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from scipy.sparse.linalg import eigsh
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from scipy.sparse.csgraph import connected_components
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from scipy.sparse.csgraph import laplacian as csgraph_laplacian
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from ..base import BaseEstimator
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from ..utils import (
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check_array,
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check_random_state,
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check_symmetric,
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)
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from ..utils._arpack import _init_arpack_v0
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from ..utils.extmath import _deterministic_vector_sign_flip
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from ..utils._param_validation import Interval, StrOptions
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from ..utils.fixes import lobpcg
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from ..metrics.pairwise import rbf_kernel
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from ..neighbors import kneighbors_graph, NearestNeighbors
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def _graph_connected_component(graph, node_id):
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"""Find the largest graph connected components that contains one
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given node.
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Parameters
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----------
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graph : array-like of shape (n_samples, n_samples)
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Adjacency matrix of the graph, non-zero weight means an edge
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between the nodes.
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node_id : int
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The index of the query node of the graph.
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Returns
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-------
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connected_components_matrix : array-like of shape (n_samples,)
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An array of bool value indicating the indexes of the nodes
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belonging to the largest connected components of the given query
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node.
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"""
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n_node = graph.shape[0]
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if sparse.issparse(graph):
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# speed up row-wise access to boolean connection mask
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graph = graph.tocsr()
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connected_nodes = np.zeros(n_node, dtype=bool)
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nodes_to_explore = np.zeros(n_node, dtype=bool)
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nodes_to_explore[node_id] = True
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for _ in range(n_node):
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last_num_component = connected_nodes.sum()
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np.logical_or(connected_nodes, nodes_to_explore, out=connected_nodes)
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if last_num_component >= connected_nodes.sum():
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break
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indices = np.where(nodes_to_explore)[0]
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nodes_to_explore.fill(False)
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for i in indices:
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if sparse.issparse(graph):
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neighbors = graph[i].toarray().ravel()
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else:
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neighbors = graph[i]
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np.logical_or(nodes_to_explore, neighbors, out=nodes_to_explore)
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return connected_nodes
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def _graph_is_connected(graph):
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"""Return whether the graph is connected (True) or Not (False).
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Parameters
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----------
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graph : {array-like, sparse matrix} of shape (n_samples, n_samples)
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Adjacency matrix of the graph, non-zero weight means an edge
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between the nodes.
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Returns
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-------
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is_connected : bool
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True means the graph is fully connected and False means not.
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"""
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if sparse.isspmatrix(graph):
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# sparse graph, find all the connected components
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n_connected_components, _ = connected_components(graph)
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return n_connected_components == 1
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else:
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# dense graph, find all connected components start from node 0
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return _graph_connected_component(graph, 0).sum() == graph.shape[0]
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def _set_diag(laplacian, value, norm_laplacian):
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"""Set the diagonal of the laplacian matrix and convert it to a
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sparse format well suited for eigenvalue decomposition.
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Parameters
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----------
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laplacian : {ndarray, sparse matrix}
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The graph laplacian.
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value : float
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The value of the diagonal.
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norm_laplacian : bool
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Whether the value of the diagonal should be changed or not.
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Returns
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-------
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laplacian : {array, sparse matrix}
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An array of matrix in a form that is well suited to fast
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eigenvalue decomposition, depending on the band width of the
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matrix.
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"""
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n_nodes = laplacian.shape[0]
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# We need all entries in the diagonal to values
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if not sparse.isspmatrix(laplacian):
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if norm_laplacian:
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laplacian.flat[:: n_nodes + 1] = value
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else:
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laplacian = laplacian.tocoo()
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if norm_laplacian:
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diag_idx = laplacian.row == laplacian.col
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laplacian.data[diag_idx] = value
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# If the matrix has a small number of diagonals (as in the
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# case of structured matrices coming from images), the
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# dia format might be best suited for matvec products:
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n_diags = np.unique(laplacian.row - laplacian.col).size
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if n_diags <= 7:
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# 3 or less outer diagonals on each side
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laplacian = laplacian.todia()
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else:
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# csr has the fastest matvec and is thus best suited to
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# arpack
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laplacian = laplacian.tocsr()
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return laplacian
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def spectral_embedding(
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adjacency,
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*,
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n_components=8,
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eigen_solver=None,
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random_state=None,
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eigen_tol="auto",
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norm_laplacian=True,
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drop_first=True,
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):
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"""Project the sample on the first eigenvectors of the graph Laplacian.
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The adjacency matrix is used to compute a normalized graph Laplacian
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whose spectrum (especially the eigenvectors associated to the
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smallest eigenvalues) has an interpretation in terms of minimal
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number of cuts necessary to split the graph into comparably sized
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components.
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This embedding can also 'work' even if the ``adjacency`` variable is
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not strictly the adjacency matrix of a graph but more generally
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an affinity or similarity matrix between samples (for instance the
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heat kernel of a euclidean distance matrix or a k-NN matrix).
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However care must taken to always make the affinity matrix symmetric
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so that the eigenvector decomposition works as expected.
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Note : Laplacian Eigenmaps is the actual algorithm implemented here.
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Read more in the :ref:`User Guide <spectral_embedding>`.
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Parameters
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----------
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adjacency : {array-like, sparse graph} of shape (n_samples, n_samples)
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The adjacency matrix of the graph to embed.
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n_components : int, default=8
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The dimension of the projection subspace.
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eigen_solver : {'arpack', 'lobpcg', 'amg'}, default=None
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The eigenvalue decomposition strategy to use. AMG requires pyamg
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to be installed. It can be faster on very large, sparse problems,
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but may also lead to instabilities. If None, then ``'arpack'`` is
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used.
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random_state : int, RandomState instance or None, default=None
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A pseudo random number generator used for the initialization
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of the lobpcg eigen vectors decomposition when `eigen_solver ==
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'amg'`, and for the K-Means initialization. Use an int to make
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the results deterministic across calls (See
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:term:`Glossary <random_state>`).
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.. note::
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When using `eigen_solver == 'amg'`,
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it is necessary to also fix the global numpy seed with
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`np.random.seed(int)` to get deterministic results. See
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https://github.com/pyamg/pyamg/issues/139 for further
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information.
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eigen_tol : float, default="auto"
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Stopping criterion for eigendecomposition of the Laplacian matrix.
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If `eigen_tol="auto"` then the passed tolerance will depend on the
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`eigen_solver`:
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- If `eigen_solver="arpack"`, then `eigen_tol=0.0`;
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- If `eigen_solver="lobpcg"` or `eigen_solver="amg"`, then
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`eigen_tol=None` which configures the underlying `lobpcg` solver to
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automatically resolve the value according to their heuristics. See,
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:func:`scipy.sparse.linalg.lobpcg` for details.
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Note that when using `eigen_solver="amg"` values of `tol<1e-5` may lead
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to convergence issues and should be avoided.
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.. versionadded:: 1.2
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Added 'auto' option.
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norm_laplacian : bool, default=True
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If True, then compute symmetric normalized Laplacian.
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drop_first : bool, default=True
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Whether to drop the first eigenvector. For spectral embedding, this
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should be True as the first eigenvector should be constant vector for
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connected graph, but for spectral clustering, this should be kept as
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False to retain the first eigenvector.
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Returns
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-------
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embedding : ndarray of shape (n_samples, n_components)
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The reduced samples.
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Notes
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-----
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Spectral Embedding (Laplacian Eigenmaps) is most useful when the graph
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has one connected component. If there graph has many components, the first
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few eigenvectors will simply uncover the connected components of the graph.
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References
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----------
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* https://en.wikipedia.org/wiki/LOBPCG
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* :doi:`"Toward the Optimal Preconditioned Eigensolver: Locally Optimal
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Block Preconditioned Conjugate Gradient Method",
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Andrew V. Knyazev
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<10.1137/S1064827500366124>`
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"""
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adjacency = check_symmetric(adjacency)
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try:
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from pyamg import smoothed_aggregation_solver
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except ImportError as e:
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if eigen_solver == "amg":
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raise ValueError(
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"The eigen_solver was set to 'amg', but pyamg is not available."
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) from e
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if eigen_solver is None:
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eigen_solver = "arpack"
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elif eigen_solver not in ("arpack", "lobpcg", "amg"):
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raise ValueError(
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"Unknown value for eigen_solver: '%s'."
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"Should be 'amg', 'arpack', or 'lobpcg'" % eigen_solver
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)
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random_state = check_random_state(random_state)
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n_nodes = adjacency.shape[0]
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# Whether to drop the first eigenvector
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if drop_first:
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n_components = n_components + 1
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if not _graph_is_connected(adjacency):
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warnings.warn(
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"Graph is not fully connected, spectral embedding may not work as expected."
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)
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laplacian, dd = csgraph_laplacian(
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adjacency, normed=norm_laplacian, return_diag=True
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)
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if (
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eigen_solver == "arpack"
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or eigen_solver != "lobpcg"
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and (not sparse.isspmatrix(laplacian) or n_nodes < 5 * n_components)
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):
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# lobpcg used with eigen_solver='amg' has bugs for low number of nodes
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# for details see the source code in scipy:
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# https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen
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# /lobpcg/lobpcg.py#L237
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# or matlab:
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# https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m
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laplacian = _set_diag(laplacian, 1, norm_laplacian)
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# Here we'll use shift-invert mode for fast eigenvalues
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# (see https://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
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# for a short explanation of what this means)
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# Because the normalized Laplacian has eigenvalues between 0 and 2,
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# I - L has eigenvalues between -1 and 1. ARPACK is most efficient
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# when finding eigenvalues of largest magnitude (keyword which='LM')
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# and when these eigenvalues are very large compared to the rest.
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# For very large, very sparse graphs, I - L can have many, many
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# eigenvalues very near 1.0. This leads to slow convergence. So
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# instead, we'll use ARPACK's shift-invert mode, asking for the
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# eigenvalues near 1.0. This effectively spreads-out the spectrum
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# near 1.0 and leads to much faster convergence: potentially an
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# orders-of-magnitude speedup over simply using keyword which='LA'
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# in standard mode.
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try:
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# We are computing the opposite of the laplacian inplace so as
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# to spare a memory allocation of a possibly very large array
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tol = 0 if eigen_tol == "auto" else eigen_tol
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laplacian *= -1
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v0 = _init_arpack_v0(laplacian.shape[0], random_state)
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_, diffusion_map = eigsh(
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laplacian, k=n_components, sigma=1.0, which="LM", tol=tol, v0=v0
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)
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embedding = diffusion_map.T[n_components::-1]
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if norm_laplacian:
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# recover u = D^-1/2 x from the eigenvector output x
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embedding = embedding / dd
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except RuntimeError:
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# When submatrices are exactly singular, an LU decomposition
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# in arpack fails. We fallback to lobpcg
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eigen_solver = "lobpcg"
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# Revert the laplacian to its opposite to have lobpcg work
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laplacian *= -1
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elif eigen_solver == "amg":
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# Use AMG to get a preconditioner and speed up the eigenvalue
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# problem.
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if not sparse.issparse(laplacian):
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warnings.warn("AMG works better for sparse matrices")
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laplacian = check_array(
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laplacian, dtype=[np.float64, np.float32], accept_sparse=True
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)
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laplacian = _set_diag(laplacian, 1, norm_laplacian)
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# The Laplacian matrix is always singular, having at least one zero
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# eigenvalue, corresponding to the trivial eigenvector, which is a
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# constant. Using a singular matrix for preconditioning may result in
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# random failures in LOBPCG and is not supported by the existing
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# theory:
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# see https://doi.org/10.1007/s10208-015-9297-1
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# Shift the Laplacian so its diagononal is not all ones. The shift
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# does change the eigenpairs however, so we'll feed the shifted
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# matrix to the solver and afterward set it back to the original.
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diag_shift = 1e-5 * sparse.eye(laplacian.shape[0])
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laplacian += diag_shift
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ml = smoothed_aggregation_solver(check_array(laplacian, accept_sparse="csr"))
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laplacian -= diag_shift
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M = ml.aspreconditioner()
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# Create initial approximation X to eigenvectors
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X = random_state.standard_normal(size=(laplacian.shape[0], n_components + 1))
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X[:, 0] = dd.ravel()
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X = X.astype(laplacian.dtype)
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tol = None if eigen_tol == "auto" else eigen_tol
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_, diffusion_map = lobpcg(laplacian, X, M=M, tol=tol, largest=False)
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embedding = diffusion_map.T
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if norm_laplacian:
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# recover u = D^-1/2 x from the eigenvector output x
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embedding = embedding / dd
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if embedding.shape[0] == 1:
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raise ValueError
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if eigen_solver == "lobpcg":
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laplacian = check_array(
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laplacian, dtype=[np.float64, np.float32], accept_sparse=True
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)
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if n_nodes < 5 * n_components + 1:
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# see note above under arpack why lobpcg has problems with small
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# number of nodes
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# lobpcg will fallback to eigh, so we short circuit it
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if sparse.isspmatrix(laplacian):
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laplacian = laplacian.toarray()
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_, diffusion_map = eigh(laplacian, check_finite=False)
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embedding = diffusion_map.T[:n_components]
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if norm_laplacian:
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# recover u = D^-1/2 x from the eigenvector output x
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embedding = embedding / dd
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else:
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laplacian = _set_diag(laplacian, 1, norm_laplacian)
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# We increase the number of eigenvectors requested, as lobpcg
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# doesn't behave well in low dimension and create initial
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# approximation X to eigenvectors
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X = random_state.standard_normal(
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size=(laplacian.shape[0], n_components + 1)
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)
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X[:, 0] = dd.ravel()
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X = X.astype(laplacian.dtype)
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tol = None if eigen_tol == "auto" else eigen_tol
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_, diffusion_map = lobpcg(
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laplacian, X, tol=tol, largest=False, maxiter=2000
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)
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embedding = diffusion_map.T[:n_components]
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if norm_laplacian:
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# recover u = D^-1/2 x from the eigenvector output x
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embedding = embedding / dd
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if embedding.shape[0] == 1:
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raise ValueError
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embedding = _deterministic_vector_sign_flip(embedding)
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if drop_first:
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return embedding[1:n_components].T
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else:
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return embedding[:n_components].T
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class SpectralEmbedding(BaseEstimator):
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"""Spectral embedding for non-linear dimensionality reduction.
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Forms an affinity matrix given by the specified function and
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applies spectral decomposition to the corresponding graph laplacian.
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The resulting transformation is given by the value of the
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eigenvectors for each data point.
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Note : Laplacian Eigenmaps is the actual algorithm implemented here.
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Read more in the :ref:`User Guide <spectral_embedding>`.
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Parameters
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----------
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n_components : int, default=2
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The dimension of the projected subspace.
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affinity : {'nearest_neighbors', 'rbf', 'precomputed', \
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'precomputed_nearest_neighbors'} or callable, \
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default='nearest_neighbors'
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How to construct the affinity matrix.
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- 'nearest_neighbors' : construct the affinity matrix by computing a
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graph of nearest neighbors.
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- 'rbf' : construct the affinity matrix by computing a radial basis
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function (RBF) kernel.
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- 'precomputed' : interpret ``X`` as a precomputed affinity matrix.
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- 'precomputed_nearest_neighbors' : interpret ``X`` as a sparse graph
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of precomputed nearest neighbors, and constructs the affinity matrix
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by selecting the ``n_neighbors`` nearest neighbors.
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- callable : use passed in function as affinity
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the function takes in data matrix (n_samples, n_features)
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and return affinity matrix (n_samples, n_samples).
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gamma : float, default=None
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Kernel coefficient for rbf kernel. If None, gamma will be set to
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1/n_features.
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random_state : int, RandomState instance or None, default=None
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A pseudo random number generator used for the initialization
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of the lobpcg eigen vectors decomposition when `eigen_solver ==
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'amg'`, and for the K-Means initialization. Use an int to make
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the results deterministic across calls (See
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:term:`Glossary <random_state>`).
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.. note::
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|
When using `eigen_solver == 'amg'`,
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|
it is necessary to also fix the global numpy seed with
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`np.random.seed(int)` to get deterministic results. See
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|
https://github.com/pyamg/pyamg/issues/139 for further
|
|
information.
|
|
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eigen_solver : {'arpack', 'lobpcg', 'amg'}, default=None
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The eigenvalue decomposition strategy to use. AMG requires pyamg
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to be installed. It can be faster on very large, sparse problems.
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If None, then ``'arpack'`` is used.
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eigen_tol : float, default="auto"
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Stopping criterion for eigendecomposition of the Laplacian matrix.
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If `eigen_tol="auto"` then the passed tolerance will depend on the
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`eigen_solver`:
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- If `eigen_solver="arpack"`, then `eigen_tol=0.0`;
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- If `eigen_solver="lobpcg"` or `eigen_solver="amg"`, then
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`eigen_tol=None` which configures the underlying `lobpcg` solver to
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automatically resolve the value according to their heuristics. See,
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:func:`scipy.sparse.linalg.lobpcg` for details.
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Note that when using `eigen_solver="lobpcg"` or `eigen_solver="amg"`
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values of `tol<1e-5` may lead to convergence issues and should be
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|
avoided.
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.. versionadded:: 1.2
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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 <n_jobs>`
|
|
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
|
|
<https://citeseerx.ist.psu.edu/doc_view/pid/796c5d6336fc52aa84db575fb821c78918b65f58>`_
|
|
|
|
- :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 _more_tags(self):
|
|
return {
|
|
"pairwise": self.affinity
|
|
in ["precomputed", "precomputed_nearest_neighbors"]
|
|
}
|
|
|
|
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_
|
|
|
|
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.
|
|
"""
|
|
self._validate_params()
|
|
|
|
X = self._validate_data(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_
|