553 lines
22 KiB
Python
553 lines
22 KiB
Python
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# -*- coding: utf-8 -*-
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"""Algorithms for spectral clustering"""
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# Author: Gael Varoquaux gael.varoquaux@normalesup.org
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# Brian Cheung
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# Wei LI <kuantkid@gmail.com>
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# License: BSD 3 clause
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import warnings
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import numpy as np
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from ..base import BaseEstimator, ClusterMixin
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from ..utils import check_random_state, as_float_array
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from ..utils.validation import _deprecate_positional_args
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from ..metrics.pairwise import pairwise_kernels
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from ..neighbors import kneighbors_graph, NearestNeighbors
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from ..manifold import spectral_embedding
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from ._kmeans import k_means
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@_deprecate_positional_args
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def discretize(vectors, *, copy=True, max_svd_restarts=30, n_iter_max=20,
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random_state=None):
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"""Search for a partition matrix (clustering) which is closest to the
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eigenvector embedding.
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Parameters
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----------
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vectors : array-like, shape: (n_samples, n_clusters)
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The embedding space of the samples.
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copy : boolean, optional, default: True
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Whether to copy vectors, or perform in-place normalization.
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max_svd_restarts : int, optional, default: 30
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Maximum number of attempts to restart SVD if convergence fails
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n_iter_max : int, optional, default: 30
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Maximum number of iterations to attempt in rotation and partition
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matrix search if machine precision convergence is not reached
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random_state : int, RandomState instance, default=None
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Determines random number generation for rotation matrix initialization.
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Use an int to make the randomness deterministic.
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See :term:`Glossary <random_state>`.
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Returns
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-------
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labels : array of integers, shape: n_samples
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The labels of the clusters.
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References
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----------
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- Multiclass spectral clustering, 2003
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Stella X. Yu, Jianbo Shi
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https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
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Notes
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-----
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The eigenvector embedding is used to iteratively search for the
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closest discrete partition. First, the eigenvector embedding is
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normalized to the space of partition matrices. An optimal discrete
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partition matrix closest to this normalized embedding multiplied by
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an initial rotation is calculated. Fixing this discrete partition
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matrix, an optimal rotation matrix is calculated. These two
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calculations are performed until convergence. The discrete partition
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matrix is returned as the clustering solution. Used in spectral
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clustering, this method tends to be faster and more robust to random
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initialization than k-means.
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"""
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from scipy.sparse import csc_matrix
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from scipy.linalg import LinAlgError
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random_state = check_random_state(random_state)
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vectors = as_float_array(vectors, copy=copy)
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eps = np.finfo(float).eps
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n_samples, n_components = vectors.shape
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# Normalize the eigenvectors to an equal length of a vector of ones.
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# Reorient the eigenvectors to point in the negative direction with respect
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# to the first element. This may have to do with constraining the
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# eigenvectors to lie in a specific quadrant to make the discretization
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# search easier.
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norm_ones = np.sqrt(n_samples)
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for i in range(vectors.shape[1]):
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vectors[:, i] = (vectors[:, i] / np.linalg.norm(vectors[:, i])) \
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* norm_ones
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if vectors[0, i] != 0:
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vectors[:, i] = -1 * vectors[:, i] * np.sign(vectors[0, i])
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# Normalize the rows of the eigenvectors. Samples should lie on the unit
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# hypersphere centered at the origin. This transforms the samples in the
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# embedding space to the space of partition matrices.
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vectors = vectors / np.sqrt((vectors ** 2).sum(axis=1))[:, np.newaxis]
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svd_restarts = 0
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has_converged = False
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# If there is an exception we try to randomize and rerun SVD again
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# do this max_svd_restarts times.
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while (svd_restarts < max_svd_restarts) and not has_converged:
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# Initialize first column of rotation matrix with a row of the
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# eigenvectors
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rotation = np.zeros((n_components, n_components))
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rotation[:, 0] = vectors[random_state.randint(n_samples), :].T
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# To initialize the rest of the rotation matrix, find the rows
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# of the eigenvectors that are as orthogonal to each other as
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# possible
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c = np.zeros(n_samples)
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for j in range(1, n_components):
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# Accumulate c to ensure row is as orthogonal as possible to
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# previous picks as well as current one
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c += np.abs(np.dot(vectors, rotation[:, j - 1]))
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rotation[:, j] = vectors[c.argmin(), :].T
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last_objective_value = 0.0
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n_iter = 0
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while not has_converged:
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n_iter += 1
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t_discrete = np.dot(vectors, rotation)
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labels = t_discrete.argmax(axis=1)
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vectors_discrete = csc_matrix(
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(np.ones(len(labels)), (np.arange(0, n_samples), labels)),
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shape=(n_samples, n_components))
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t_svd = vectors_discrete.T * vectors
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try:
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U, S, Vh = np.linalg.svd(t_svd)
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svd_restarts += 1
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except LinAlgError:
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print("SVD did not converge, randomizing and trying again")
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break
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ncut_value = 2.0 * (n_samples - S.sum())
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if ((abs(ncut_value - last_objective_value) < eps) or
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(n_iter > n_iter_max)):
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has_converged = True
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else:
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# otherwise calculate rotation and continue
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last_objective_value = ncut_value
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rotation = np.dot(Vh.T, U.T)
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if not has_converged:
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raise LinAlgError('SVD did not converge')
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return labels
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@_deprecate_positional_args
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def spectral_clustering(affinity, *, n_clusters=8, n_components=None,
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eigen_solver=None, random_state=None, n_init=10,
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eigen_tol=0.0, assign_labels='kmeans'):
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"""Apply clustering to a projection of the normalized Laplacian.
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In practice Spectral Clustering is very useful when the structure of
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the individual clusters is highly non-convex or more generally when
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a measure of the center and spread of the cluster is not a suitable
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description of the complete cluster. For instance, when clusters are
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nested circles on the 2D plane.
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If affinity is the adjacency matrix of a graph, this method can be
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used to find normalized graph cuts.
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Read more in the :ref:`User Guide <spectral_clustering>`.
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Parameters
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----------
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affinity : array-like or sparse matrix, shape: (n_samples, n_samples)
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The affinity matrix describing the relationship of the samples to
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embed. **Must be symmetric**.
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Possible examples:
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- adjacency matrix of a graph,
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- heat kernel of the pairwise distance matrix of the samples,
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- symmetric k-nearest neighbours connectivity matrix of the samples.
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n_clusters : integer, optional
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Number of clusters to extract.
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n_components : integer, optional, default is n_clusters
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Number of eigen vectors to use for the spectral embedding
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eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}
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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
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random_state : int, RandomState instance, default=None
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A pseudo random number generator used for the initialization of the
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lobpcg eigen vectors decomposition when eigen_solver == 'amg' and by
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the K-Means initialization. Use an int to make the randomness
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deterministic.
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See :term:`Glossary <random_state>`.
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n_init : int, optional, default: 10
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Number of time the k-means algorithm will be run with different
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centroid seeds. The final results will be the best output of
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n_init consecutive runs in terms of inertia.
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eigen_tol : float, optional, default: 0.0
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Stopping criterion for eigendecomposition of the Laplacian matrix
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when using arpack eigen_solver.
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assign_labels : {'kmeans', 'discretize'}, default: 'kmeans'
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The strategy to use to assign labels in the embedding
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space. There are two ways to assign labels after the laplacian
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embedding. k-means can be applied and is a popular choice. But it can
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also be sensitive to initialization. Discretization is another
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approach which is less sensitive to random initialization. See
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the 'Multiclass spectral clustering' paper referenced below for
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more details on the discretization approach.
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Returns
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-------
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labels : array of integers, shape: n_samples
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The labels of the clusters.
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References
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----------
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- Normalized cuts and image segmentation, 2000
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Jianbo Shi, Jitendra Malik
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http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
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- A Tutorial on Spectral Clustering, 2007
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Ulrike von Luxburg
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
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- Multiclass spectral clustering, 2003
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Stella X. Yu, Jianbo Shi
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https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
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Notes
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-----
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The graph should contain only one connect component, elsewhere
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the results make little sense.
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This algorithm solves the normalized cut for k=2: it is a
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normalized spectral clustering.
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"""
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if assign_labels not in ('kmeans', 'discretize'):
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raise ValueError("The 'assign_labels' parameter should be "
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"'kmeans' or 'discretize', but '%s' was given"
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% assign_labels)
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random_state = check_random_state(random_state)
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n_components = n_clusters if n_components is None else n_components
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# The first eigen vector is constant only for fully connected graphs
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# and should be kept for spectral clustering (drop_first = False)
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# See spectral_embedding documentation.
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maps = spectral_embedding(affinity, n_components=n_components,
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eigen_solver=eigen_solver,
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random_state=random_state,
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eigen_tol=eigen_tol, drop_first=False)
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if assign_labels == 'kmeans':
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_, labels, _ = k_means(maps, n_clusters, random_state=random_state,
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n_init=n_init)
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else:
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labels = discretize(maps, random_state=random_state)
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return labels
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class SpectralClustering(ClusterMixin, BaseEstimator):
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"""Apply clustering to a projection of the normalized Laplacian.
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In practice Spectral Clustering is very useful when the structure of
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the individual clusters is highly non-convex or more generally when
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a measure of the center and spread of the cluster is not a suitable
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description of the complete cluster. For instance when clusters are
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nested circles on the 2D plane.
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If affinity is the adjacency matrix of a graph, this method can be
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used to find normalized graph cuts.
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When calling ``fit``, an affinity matrix is constructed using either
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kernel function such the Gaussian (aka RBF) kernel of the euclidean
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distanced ``d(X, X)``::
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np.exp(-gamma * d(X,X) ** 2)
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or a k-nearest neighbors connectivity matrix.
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Alternatively, using ``precomputed``, a user-provided affinity
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matrix can be used.
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Read more in the :ref:`User Guide <spectral_clustering>`.
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Parameters
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----------
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n_clusters : integer, optional
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The dimension of the projection subspace.
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eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}
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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.
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n_components : integer, optional, default=n_clusters
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Number of eigen vectors to use for the spectral embedding
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random_state : int, RandomState instance, default=None
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A pseudo random number generator used for the initialization of the
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lobpcg eigen vectors decomposition when ``eigen_solver='amg'`` and by
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the K-Means initialization. Use an int to make the randomness
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deterministic.
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See :term:`Glossary <random_state>`.
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n_init : int, optional, default: 10
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Number of time the k-means algorithm will be run with different
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centroid seeds. The final results will be the best output of
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n_init consecutive runs in terms of inertia.
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gamma : float, default=1.0
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Kernel coefficient for rbf, poly, sigmoid, laplacian and chi2 kernels.
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Ignored for ``affinity='nearest_neighbors'``.
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affinity : string or callable, default 'rbf'
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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 using a radial basis function
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(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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- one of the kernels supported by
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:func:`~sklearn.metrics.pairwise_kernels`.
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Only kernels that produce similarity scores (non-negative values that
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increase with similarity) should be used. This property is not checked
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by the clustering algorithm.
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n_neighbors : integer
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Number of neighbors to use when constructing the affinity matrix using
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the nearest neighbors method. Ignored for ``affinity='rbf'``.
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eigen_tol : float, optional, default: 0.0
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Stopping criterion for eigendecomposition of the Laplacian matrix
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when ``eigen_solver='arpack'``.
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assign_labels : {'kmeans', 'discretize'}, default: 'kmeans'
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The strategy to use to assign labels in the embedding
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space. There are two ways to assign labels after the laplacian
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embedding. k-means can be applied and is a popular choice. But it can
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also be sensitive to initialization. Discretization is another approach
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which is less sensitive to random initialization.
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degree : float, default=3
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Degree of the polynomial kernel. Ignored by other kernels.
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coef0 : float, default=1
|
||
|
|
Zero coefficient for polynomial and sigmoid kernels.
|
||
|
|
Ignored by other kernels.
|
||
|
|
|
||
|
|
kernel_params : dictionary of string to any, optional
|
||
|
|
Parameters (keyword arguments) and values for kernel passed as
|
||
|
|
callable object. Ignored by other kernels.
|
||
|
|
|
||
|
|
n_jobs : int or None, optional (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
|
||
|
|
----------
|
||
|
|
affinity_matrix_ : array-like, shape (n_samples, n_samples)
|
||
|
|
Affinity matrix used for clustering. Available only if after calling
|
||
|
|
``fit``.
|
||
|
|
|
||
|
|
labels_ : array, shape (n_samples,)
|
||
|
|
Labels of each point
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from sklearn.cluster import SpectralClustering
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> X = np.array([[1, 1], [2, 1], [1, 0],
|
||
|
|
... [4, 7], [3, 5], [3, 6]])
|
||
|
|
>>> clustering = SpectralClustering(n_clusters=2,
|
||
|
|
... assign_labels="discretize",
|
||
|
|
... random_state=0).fit(X)
|
||
|
|
>>> clustering.labels_
|
||
|
|
array([1, 1, 1, 0, 0, 0])
|
||
|
|
>>> clustering
|
||
|
|
SpectralClustering(assign_labels='discretize', n_clusters=2,
|
||
|
|
random_state=0)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
If you have an affinity matrix, such as a distance matrix,
|
||
|
|
for which 0 means identical elements, and high values means
|
||
|
|
very dissimilar elements, it can be transformed in a
|
||
|
|
similarity matrix that is well suited for the algorithm by
|
||
|
|
applying the Gaussian (RBF, heat) kernel::
|
||
|
|
|
||
|
|
np.exp(- dist_matrix ** 2 / (2. * delta ** 2))
|
||
|
|
|
||
|
|
Where ``delta`` is a free parameter representing the width of the Gaussian
|
||
|
|
kernel.
|
||
|
|
|
||
|
|
Another alternative is to take a symmetric version of the k
|
||
|
|
nearest neighbors connectivity matrix of the points.
|
||
|
|
|
||
|
|
If the pyamg package is installed, it is used: this greatly
|
||
|
|
speeds up computation.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
- Normalized cuts and image segmentation, 2000
|
||
|
|
Jianbo Shi, Jitendra Malik
|
||
|
|
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
|
||
|
|
|
||
|
|
- A Tutorial on Spectral Clustering, 2007
|
||
|
|
Ulrike von Luxburg
|
||
|
|
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
|
||
|
|
|
||
|
|
- Multiclass spectral clustering, 2003
|
||
|
|
Stella X. Yu, Jianbo Shi
|
||
|
|
https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def __init__(self, n_clusters=8, *, eigen_solver=None, n_components=None,
|
||
|
|
random_state=None, n_init=10, gamma=1., affinity='rbf',
|
||
|
|
n_neighbors=10, eigen_tol=0.0, assign_labels='kmeans',
|
||
|
|
degree=3, coef0=1, kernel_params=None, n_jobs=None):
|
||
|
|
self.n_clusters = n_clusters
|
||
|
|
self.eigen_solver = eigen_solver
|
||
|
|
self.n_components = n_components
|
||
|
|
self.random_state = random_state
|
||
|
|
self.n_init = n_init
|
||
|
|
self.gamma = gamma
|
||
|
|
self.affinity = affinity
|
||
|
|
self.n_neighbors = n_neighbors
|
||
|
|
self.eigen_tol = eigen_tol
|
||
|
|
self.assign_labels = assign_labels
|
||
|
|
self.degree = degree
|
||
|
|
self.coef0 = coef0
|
||
|
|
self.kernel_params = kernel_params
|
||
|
|
self.n_jobs = n_jobs
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Perform spectral clustering from features, or affinity matrix.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like or sparse matrix, shape (n_samples, n_features), or \
|
||
|
|
array-like, shape (n_samples, n_samples)
|
||
|
|
Training instances to cluster, or similarities / affinities between
|
||
|
|
instances if ``affinity='precomputed'``. If a sparse matrix is
|
||
|
|
provided in a format other than ``csr_matrix``, ``csc_matrix``,
|
||
|
|
or ``coo_matrix``, it will be converted into a sparse
|
||
|
|
``csr_matrix``.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
Not used, present here for API consistency by convention.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self
|
||
|
|
|
||
|
|
"""
|
||
|
|
X = self._validate_data(X, accept_sparse=['csr', 'csc', 'coo'],
|
||
|
|
dtype=np.float64, ensure_min_samples=2)
|
||
|
|
allow_squared = self.affinity in ["precomputed",
|
||
|
|
"precomputed_nearest_neighbors"]
|
||
|
|
if X.shape[0] == X.shape[1] and not allow_squared:
|
||
|
|
warnings.warn("The spectral clustering API has changed. ``fit``"
|
||
|
|
"now constructs an affinity matrix from data. To use"
|
||
|
|
" a custom affinity matrix, "
|
||
|
|
"set ``affinity=precomputed``.")
|
||
|
|
|
||
|
|
if self.affinity == 'nearest_neighbors':
|
||
|
|
connectivity = kneighbors_graph(X, n_neighbors=self.n_neighbors,
|
||
|
|
include_self=True,
|
||
|
|
n_jobs=self.n_jobs)
|
||
|
|
self.affinity_matrix_ = 0.5 * (connectivity + connectivity.T)
|
||
|
|
elif 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)
|
||
|
|
elif self.affinity == 'precomputed':
|
||
|
|
self.affinity_matrix_ = X
|
||
|
|
else:
|
||
|
|
params = self.kernel_params
|
||
|
|
if params is None:
|
||
|
|
params = {}
|
||
|
|
if not callable(self.affinity):
|
||
|
|
params['gamma'] = self.gamma
|
||
|
|
params['degree'] = self.degree
|
||
|
|
params['coef0'] = self.coef0
|
||
|
|
self.affinity_matrix_ = pairwise_kernels(X, metric=self.affinity,
|
||
|
|
filter_params=True,
|
||
|
|
**params)
|
||
|
|
|
||
|
|
random_state = check_random_state(self.random_state)
|
||
|
|
self.labels_ = spectral_clustering(self.affinity_matrix_,
|
||
|
|
n_clusters=self.n_clusters,
|
||
|
|
n_components=self.n_components,
|
||
|
|
eigen_solver=self.eigen_solver,
|
||
|
|
random_state=random_state,
|
||
|
|
n_init=self.n_init,
|
||
|
|
eigen_tol=self.eigen_tol,
|
||
|
|
assign_labels=self.assign_labels)
|
||
|
|
return self
|
||
|
|
|
||
|
|
def fit_predict(self, X, y=None):
|
||
|
|
"""Perform spectral clustering from features, or affinity matrix,
|
||
|
|
and return cluster labels.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like or sparse matrix, shape (n_samples, n_features), or \
|
||
|
|
array-like, shape (n_samples, n_samples)
|
||
|
|
Training instances to cluster, or similarities / affinities between
|
||
|
|
instances if ``affinity='precomputed'``. If a sparse matrix is
|
||
|
|
provided in a format other than ``csr_matrix``, ``csc_matrix``,
|
||
|
|
or ``coo_matrix``, it will be converted into a sparse
|
||
|
|
``csr_matrix``.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
Not used, present here for API consistency by convention.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
labels : ndarray, shape (n_samples,)
|
||
|
|
Cluster labels.
|
||
|
|
"""
|
||
|
|
return super().fit_predict(X, y)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def _pairwise(self):
|
||
|
|
return self.affinity in ["precomputed",
|
||
|
|
"precomputed_nearest_neighbors"]
|