735 lines
27 KiB
Python
735 lines
27 KiB
Python
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"""Locally Linear Embedding"""
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# Author: Fabian Pedregosa -- <fabian.pedregosa@inria.fr>
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# Jake Vanderplas -- <vanderplas@astro.washington.edu>
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# License: BSD 3 clause (C) INRIA 2011
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import numpy as np
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from scipy.linalg import eigh, svd, qr, solve
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from scipy.sparse import eye, csr_matrix
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from scipy.sparse.linalg import eigsh
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from ..base import BaseEstimator, TransformerMixin, _UnstableArchMixin
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from ..utils import check_random_state, check_array
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from ..utils._arpack import _init_arpack_v0
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from ..utils.extmath import stable_cumsum
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from ..utils.validation import check_is_fitted
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from ..utils.validation import FLOAT_DTYPES
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from ..utils.validation import _deprecate_positional_args
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from ..neighbors import NearestNeighbors
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def barycenter_weights(X, Y, indices, reg=1e-3):
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"""Compute barycenter weights of X from Y along the first axis
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We estimate the weights to assign to each point in Y[indices] to recover
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the point X[i]. The barycenter weights sum to 1.
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Parameters
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----------
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X : array-like, shape (n_samples, n_dim)
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Y : array-like, shape (n_samples, n_dim)
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indices : array-like, shape (n_samples, n_dim)
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Indices of the points in Y used to compute the barycenter
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reg : float, default=1e-3
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amount of regularization to add for the problem to be
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well-posed in the case of n_neighbors > n_dim
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Returns
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-------
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B : array-like, shape (n_samples, n_neighbors)
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Notes
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-----
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See developers note for more information.
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"""
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X = check_array(X, dtype=FLOAT_DTYPES)
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Y = check_array(Y, dtype=FLOAT_DTYPES)
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indices = check_array(indices, dtype=int)
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n_samples, n_neighbors = indices.shape
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assert X.shape[0] == n_samples
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B = np.empty((n_samples, n_neighbors), dtype=X.dtype)
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v = np.ones(n_neighbors, dtype=X.dtype)
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# this might raise a LinalgError if G is singular and has trace
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# zero
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for i, ind in enumerate(indices):
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A = Y[ind]
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C = A - X[i] # broadcasting
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G = np.dot(C, C.T)
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trace = np.trace(G)
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if trace > 0:
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R = reg * trace
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else:
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R = reg
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G.flat[::n_neighbors + 1] += R
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w = solve(G, v, sym_pos=True)
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B[i, :] = w / np.sum(w)
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return B
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def barycenter_kneighbors_graph(X, n_neighbors, reg=1e-3, n_jobs=None):
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"""Computes the barycenter weighted graph of k-Neighbors for points in X
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Parameters
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----------
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X : {array-like, NearestNeighbors}
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Sample data, shape = (n_samples, n_features), in the form of a
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numpy array or a NearestNeighbors object.
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n_neighbors : int
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Number of neighbors for each sample.
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reg : float, default=1e-3
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Amount of regularization when solving the least-squares
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problem. Only relevant if mode='barycenter'. If None, use the
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default.
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n_jobs : int or None, default=None
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The number of parallel jobs to run for neighbors search.
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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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Returns
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-------
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A : sparse matrix in CSR format, shape = [n_samples, n_samples]
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A[i, j] is assigned the weight of edge that connects i to j.
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See Also
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--------
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sklearn.neighbors.kneighbors_graph
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sklearn.neighbors.radius_neighbors_graph
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"""
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knn = NearestNeighbors(n_neighbors=n_neighbors + 1, n_jobs=n_jobs).fit(X)
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X = knn._fit_X
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n_samples = knn.n_samples_fit_
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ind = knn.kneighbors(X, return_distance=False)[:, 1:]
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data = barycenter_weights(X, X, ind, reg=reg)
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indptr = np.arange(0, n_samples * n_neighbors + 1, n_neighbors)
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return csr_matrix((data.ravel(), ind.ravel(), indptr),
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shape=(n_samples, n_samples))
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def null_space(M, k, k_skip=1, eigen_solver='arpack', tol=1E-6, max_iter=100,
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random_state=None):
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"""
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Find the null space of a matrix M.
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Parameters
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----------
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M : {array, matrix, sparse matrix, LinearOperator}
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Input covariance matrix: should be symmetric positive semi-definite
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k : int
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Number of eigenvalues/vectors to return
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k_skip : int, default=1
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Number of low eigenvalues to skip.
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eigen_solver : {'auto', 'arpack', 'dense'}, default='arpack'
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auto : algorithm will attempt to choose the best method for input data
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arpack : use arnoldi iteration in shift-invert mode.
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For this method, M may be a dense matrix, sparse matrix,
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or general linear operator.
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Warning: ARPACK can be unstable for some problems. It is
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best to try several random seeds in order to check results.
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dense : use standard dense matrix operations for the eigenvalue
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decomposition. For this method, M must be an array
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or matrix type. This method should be avoided for
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large problems.
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tol : float, default=1e-6
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Tolerance for 'arpack' method.
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Not used if eigen_solver=='dense'.
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max_iter : int, default=100
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Maximum number of iterations for 'arpack' method.
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Not used if eigen_solver=='dense'
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random_state : int, RandomState instance, default=None
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Determines the random number generator when ``solver`` == 'arpack'.
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Pass an int for reproducible results across multiple function calls.
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See :term: `Glossary <random_state>`.
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"""
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if eigen_solver == 'auto':
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if M.shape[0] > 200 and k + k_skip < 10:
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eigen_solver = 'arpack'
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else:
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eigen_solver = 'dense'
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if eigen_solver == 'arpack':
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v0 = _init_arpack_v0(M.shape[0], random_state)
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try:
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eigen_values, eigen_vectors = eigsh(M, k + k_skip, sigma=0.0,
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tol=tol, maxiter=max_iter,
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v0=v0)
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except RuntimeError as e:
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raise ValueError(
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"Error in determining null-space with ARPACK. Error message: "
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"'%s'. Note that eigen_solver='arpack' can fail when the "
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"weight matrix is singular or otherwise ill-behaved. In that "
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"case, eigen_solver='dense' is recommended. See online "
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"documentation for more information." % e
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) from e
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return eigen_vectors[:, k_skip:], np.sum(eigen_values[k_skip:])
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elif eigen_solver == 'dense':
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if hasattr(M, 'toarray'):
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M = M.toarray()
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eigen_values, eigen_vectors = eigh(
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M, eigvals=(k_skip, k + k_skip - 1), overwrite_a=True)
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index = np.argsort(np.abs(eigen_values))
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return eigen_vectors[:, index], np.sum(eigen_values)
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else:
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raise ValueError("Unrecognized eigen_solver '%s'" % eigen_solver)
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@_deprecate_positional_args
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def locally_linear_embedding(
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X, *, n_neighbors, n_components, reg=1e-3, eigen_solver='auto',
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tol=1e-6, max_iter=100, method='standard', hessian_tol=1E-4,
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modified_tol=1E-12, random_state=None, n_jobs=None):
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"""Perform a Locally Linear Embedding analysis on the data.
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Read more in the :ref:`User Guide <locally_linear_embedding>`.
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Parameters
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----------
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X : {array-like, NearestNeighbors}
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Sample data, shape = (n_samples, n_features), in the form of a
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numpy array or a NearestNeighbors object.
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n_neighbors : int
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number of neighbors to consider for each point.
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n_components : int
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number of coordinates for the manifold.
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reg : float, default=1e-3
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regularization constant, multiplies the trace of the local covariance
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matrix of the distances.
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eigen_solver : {'auto', 'arpack', 'dense'}, default='auto'
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auto : algorithm will attempt to choose the best method for input data
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arpack : use arnoldi iteration in shift-invert mode.
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For this method, M may be a dense matrix, sparse matrix,
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or general linear operator.
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Warning: ARPACK can be unstable for some problems. It is
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best to try several random seeds in order to check results.
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dense : use standard dense matrix operations for the eigenvalue
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decomposition. For this method, M must be an array
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or matrix type. This method should be avoided for
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large problems.
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tol : float, default=1e-6
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Tolerance for 'arpack' method
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Not used if eigen_solver=='dense'.
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max_iter : int, default=100
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maximum number of iterations for the arpack solver.
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method : {'standard', 'hessian', 'modified', 'ltsa'}, default='standard'
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standard : use the standard locally linear embedding algorithm.
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see reference [1]_
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hessian : use the Hessian eigenmap method. This method requires
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n_neighbors > n_components * (1 + (n_components + 1) / 2.
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see reference [2]_
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modified : use the modified locally linear embedding algorithm.
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see reference [3]_
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ltsa : use local tangent space alignment algorithm
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see reference [4]_
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hessian_tol : float, default=1e-4
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Tolerance for Hessian eigenmapping method.
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Only used if method == 'hessian'
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modified_tol : float, default=1e-12
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Tolerance for modified LLE method.
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Only used if method == 'modified'
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random_state : int, RandomState instance, default=None
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Determines the random number generator when ``solver`` == 'arpack'.
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Pass an int for reproducible results across multiple function calls.
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See :term: `Glossary <random_state>`.
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n_jobs : int or None, default=None
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The number of parallel jobs to run for neighbors search.
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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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Returns
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-------
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Y : array-like, shape [n_samples, n_components]
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Embedding vectors.
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squared_error : float
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Reconstruction error for the embedding vectors. Equivalent to
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``norm(Y - W Y, 'fro')**2``, where W are the reconstruction weights.
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References
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----------
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.. [1] Roweis, S. & Saul, L. Nonlinear dimensionality reduction
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by locally linear embedding. Science 290:2323 (2000).
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.. [2] Donoho, D. & Grimes, C. Hessian eigenmaps: Locally
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linear embedding techniques for high-dimensional data.
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Proc Natl Acad Sci U S A. 100:5591 (2003).
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.. [3] Zhang, Z. & Wang, J. MLLE: Modified Locally Linear
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Embedding Using Multiple Weights.
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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.382
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.. [4] Zhang, Z. & Zha, H. Principal manifolds and nonlinear
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dimensionality reduction via tangent space alignment.
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Journal of Shanghai Univ. 8:406 (2004)
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"""
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if eigen_solver not in ('auto', 'arpack', 'dense'):
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raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
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if method not in ('standard', 'hessian', 'modified', 'ltsa'):
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raise ValueError("unrecognized method '%s'" % method)
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nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1, n_jobs=n_jobs)
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nbrs.fit(X)
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X = nbrs._fit_X
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N, d_in = X.shape
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if n_components > d_in:
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raise ValueError("output dimension must be less than or equal "
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"to input dimension")
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if n_neighbors >= N:
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raise ValueError(
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"Expected n_neighbors <= n_samples, "
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" but n_samples = %d, n_neighbors = %d" %
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(N, n_neighbors)
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)
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if n_neighbors <= 0:
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raise ValueError("n_neighbors must be positive")
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M_sparse = (eigen_solver != 'dense')
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if method == 'standard':
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W = barycenter_kneighbors_graph(
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nbrs, n_neighbors=n_neighbors, reg=reg, n_jobs=n_jobs)
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# we'll compute M = (I-W)'(I-W)
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# depending on the solver, we'll do this differently
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if M_sparse:
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M = eye(*W.shape, format=W.format) - W
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M = (M.T * M).tocsr()
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else:
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M = (W.T * W - W.T - W).toarray()
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M.flat[::M.shape[0] + 1] += 1 # W = W - I = W - I
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elif method == 'hessian':
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dp = n_components * (n_components + 1) // 2
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if n_neighbors <= n_components + dp:
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raise ValueError("for method='hessian', n_neighbors must be "
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"greater than "
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"[n_components * (n_components + 3) / 2]")
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neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
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return_distance=False)
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neighbors = neighbors[:, 1:]
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Yi = np.empty((n_neighbors, 1 + n_components + dp), dtype=np.float64)
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Yi[:, 0] = 1
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M = np.zeros((N, N), dtype=np.float64)
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use_svd = (n_neighbors > d_in)
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for i in range(N):
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Gi = X[neighbors[i]]
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Gi -= Gi.mean(0)
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# build Hessian estimator
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if use_svd:
|
||
|
|
U = svd(Gi, full_matrices=0)[0]
|
||
|
|
else:
|
||
|
|
Ci = np.dot(Gi, Gi.T)
|
||
|
|
U = eigh(Ci)[1][:, ::-1]
|
||
|
|
|
||
|
|
Yi[:, 1:1 + n_components] = U[:, :n_components]
|
||
|
|
|
||
|
|
j = 1 + n_components
|
||
|
|
for k in range(n_components):
|
||
|
|
Yi[:, j:j + n_components - k] = (U[:, k:k + 1] *
|
||
|
|
U[:, k:n_components])
|
||
|
|
j += n_components - k
|
||
|
|
|
||
|
|
Q, R = qr(Yi)
|
||
|
|
|
||
|
|
w = Q[:, n_components + 1:]
|
||
|
|
S = w.sum(0)
|
||
|
|
|
||
|
|
S[np.where(abs(S) < hessian_tol)] = 1
|
||
|
|
w /= S
|
||
|
|
|
||
|
|
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
|
||
|
|
M[nbrs_x, nbrs_y] += np.dot(w, w.T)
|
||
|
|
|
||
|
|
if M_sparse:
|
||
|
|
M = csr_matrix(M)
|
||
|
|
|
||
|
|
elif method == 'modified':
|
||
|
|
if n_neighbors < n_components:
|
||
|
|
raise ValueError("modified LLE requires "
|
||
|
|
"n_neighbors >= n_components")
|
||
|
|
|
||
|
|
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
|
||
|
|
return_distance=False)
|
||
|
|
neighbors = neighbors[:, 1:]
|
||
|
|
|
||
|
|
# find the eigenvectors and eigenvalues of each local covariance
|
||
|
|
# matrix. We want V[i] to be a [n_neighbors x n_neighbors] matrix,
|
||
|
|
# where the columns are eigenvectors
|
||
|
|
V = np.zeros((N, n_neighbors, n_neighbors))
|
||
|
|
nev = min(d_in, n_neighbors)
|
||
|
|
evals = np.zeros([N, nev])
|
||
|
|
|
||
|
|
# choose the most efficient way to find the eigenvectors
|
||
|
|
use_svd = (n_neighbors > d_in)
|
||
|
|
|
||
|
|
if use_svd:
|
||
|
|
for i in range(N):
|
||
|
|
X_nbrs = X[neighbors[i]] - X[i]
|
||
|
|
V[i], evals[i], _ = svd(X_nbrs,
|
||
|
|
full_matrices=True)
|
||
|
|
evals **= 2
|
||
|
|
else:
|
||
|
|
for i in range(N):
|
||
|
|
X_nbrs = X[neighbors[i]] - X[i]
|
||
|
|
C_nbrs = np.dot(X_nbrs, X_nbrs.T)
|
||
|
|
evi, vi = eigh(C_nbrs)
|
||
|
|
evals[i] = evi[::-1]
|
||
|
|
V[i] = vi[:, ::-1]
|
||
|
|
|
||
|
|
# find regularized weights: this is like normal LLE.
|
||
|
|
# because we've already computed the SVD of each covariance matrix,
|
||
|
|
# it's faster to use this rather than np.linalg.solve
|
||
|
|
reg = 1E-3 * evals.sum(1)
|
||
|
|
|
||
|
|
tmp = np.dot(V.transpose(0, 2, 1), np.ones(n_neighbors))
|
||
|
|
tmp[:, :nev] /= evals + reg[:, None]
|
||
|
|
tmp[:, nev:] /= reg[:, None]
|
||
|
|
|
||
|
|
w_reg = np.zeros((N, n_neighbors))
|
||
|
|
for i in range(N):
|
||
|
|
w_reg[i] = np.dot(V[i], tmp[i])
|
||
|
|
w_reg /= w_reg.sum(1)[:, None]
|
||
|
|
|
||
|
|
# calculate eta: the median of the ratio of small to large eigenvalues
|
||
|
|
# across the points. This is used to determine s_i, below
|
||
|
|
rho = evals[:, n_components:].sum(1) / evals[:, :n_components].sum(1)
|
||
|
|
eta = np.median(rho)
|
||
|
|
|
||
|
|
# find s_i, the size of the "almost null space" for each point:
|
||
|
|
# this is the size of the largest set of eigenvalues
|
||
|
|
# such that Sum[v; v in set]/Sum[v; v not in set] < eta
|
||
|
|
s_range = np.zeros(N, dtype=int)
|
||
|
|
evals_cumsum = stable_cumsum(evals, 1)
|
||
|
|
eta_range = evals_cumsum[:, -1:] / evals_cumsum[:, :-1] - 1
|
||
|
|
for i in range(N):
|
||
|
|
s_range[i] = np.searchsorted(eta_range[i, ::-1], eta)
|
||
|
|
s_range += n_neighbors - nev # number of zero eigenvalues
|
||
|
|
|
||
|
|
# Now calculate M.
|
||
|
|
# This is the [N x N] matrix whose null space is the desired embedding
|
||
|
|
M = np.zeros((N, N), dtype=np.float64)
|
||
|
|
for i in range(N):
|
||
|
|
s_i = s_range[i]
|
||
|
|
|
||
|
|
# select bottom s_i eigenvectors and calculate alpha
|
||
|
|
Vi = V[i, :, n_neighbors - s_i:]
|
||
|
|
alpha_i = np.linalg.norm(Vi.sum(0)) / np.sqrt(s_i)
|
||
|
|
|
||
|
|
# compute Householder matrix which satisfies
|
||
|
|
# Hi*Vi.T*ones(n_neighbors) = alpha_i*ones(s)
|
||
|
|
# using prescription from paper
|
||
|
|
h = np.full(s_i, alpha_i) - np.dot(Vi.T, np.ones(n_neighbors))
|
||
|
|
|
||
|
|
norm_h = np.linalg.norm(h)
|
||
|
|
if norm_h < modified_tol:
|
||
|
|
h *= 0
|
||
|
|
else:
|
||
|
|
h /= norm_h
|
||
|
|
|
||
|
|
# Householder matrix is
|
||
|
|
# >> Hi = np.identity(s_i) - 2*np.outer(h,h)
|
||
|
|
# Then the weight matrix is
|
||
|
|
# >> Wi = np.dot(Vi,Hi) + (1-alpha_i) * w_reg[i,:,None]
|
||
|
|
# We do this much more efficiently:
|
||
|
|
Wi = (Vi - 2 * np.outer(np.dot(Vi, h), h) +
|
||
|
|
(1 - alpha_i) * w_reg[i, :, None])
|
||
|
|
|
||
|
|
# Update M as follows:
|
||
|
|
# >> W_hat = np.zeros( (N,s_i) )
|
||
|
|
# >> W_hat[neighbors[i],:] = Wi
|
||
|
|
# >> W_hat[i] -= 1
|
||
|
|
# >> M += np.dot(W_hat,W_hat.T)
|
||
|
|
# We can do this much more efficiently:
|
||
|
|
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
|
||
|
|
M[nbrs_x, nbrs_y] += np.dot(Wi, Wi.T)
|
||
|
|
Wi_sum1 = Wi.sum(1)
|
||
|
|
M[i, neighbors[i]] -= Wi_sum1
|
||
|
|
M[neighbors[i], i] -= Wi_sum1
|
||
|
|
M[i, i] += s_i
|
||
|
|
|
||
|
|
if M_sparse:
|
||
|
|
M = csr_matrix(M)
|
||
|
|
|
||
|
|
elif method == 'ltsa':
|
||
|
|
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
|
||
|
|
return_distance=False)
|
||
|
|
neighbors = neighbors[:, 1:]
|
||
|
|
|
||
|
|
M = np.zeros((N, N))
|
||
|
|
|
||
|
|
use_svd = (n_neighbors > d_in)
|
||
|
|
|
||
|
|
for i in range(N):
|
||
|
|
Xi = X[neighbors[i]]
|
||
|
|
Xi -= Xi.mean(0)
|
||
|
|
|
||
|
|
# compute n_components largest eigenvalues of Xi * Xi^T
|
||
|
|
if use_svd:
|
||
|
|
v = svd(Xi, full_matrices=True)[0]
|
||
|
|
else:
|
||
|
|
Ci = np.dot(Xi, Xi.T)
|
||
|
|
v = eigh(Ci)[1][:, ::-1]
|
||
|
|
|
||
|
|
Gi = np.zeros((n_neighbors, n_components + 1))
|
||
|
|
Gi[:, 1:] = v[:, :n_components]
|
||
|
|
Gi[:, 0] = 1. / np.sqrt(n_neighbors)
|
||
|
|
|
||
|
|
GiGiT = np.dot(Gi, Gi.T)
|
||
|
|
|
||
|
|
nbrs_x, nbrs_y = np.meshgrid(neighbors[i], neighbors[i])
|
||
|
|
M[nbrs_x, nbrs_y] -= GiGiT
|
||
|
|
M[neighbors[i], neighbors[i]] += 1
|
||
|
|
|
||
|
|
return null_space(M, n_components, k_skip=1, eigen_solver=eigen_solver,
|
||
|
|
tol=tol, max_iter=max_iter, random_state=random_state)
|
||
|
|
|
||
|
|
|
||
|
|
class LocallyLinearEmbedding(TransformerMixin,
|
||
|
|
_UnstableArchMixin, BaseEstimator):
|
||
|
|
"""Locally Linear Embedding
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <locally_linear_embedding>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n_neighbors : int, default=5
|
||
|
|
number of neighbors to consider for each point.
|
||
|
|
|
||
|
|
n_components : int, default=2
|
||
|
|
number of coordinates for the manifold
|
||
|
|
|
||
|
|
reg : float, default=1e-3
|
||
|
|
regularization constant, multiplies the trace of the local covariance
|
||
|
|
matrix of the distances.
|
||
|
|
|
||
|
|
eigen_solver : {'auto', 'arpack', 'dense'}, default='auto'
|
||
|
|
auto : algorithm will attempt to choose the best method for input data
|
||
|
|
|
||
|
|
arpack : use arnoldi iteration in shift-invert mode.
|
||
|
|
For this method, M may be a dense matrix, sparse matrix,
|
||
|
|
or general linear operator.
|
||
|
|
Warning: ARPACK can be unstable for some problems. It is
|
||
|
|
best to try several random seeds in order to check results.
|
||
|
|
|
||
|
|
dense : use standard dense matrix operations for the eigenvalue
|
||
|
|
decomposition. For this method, M must be an array
|
||
|
|
or matrix type. This method should be avoided for
|
||
|
|
large problems.
|
||
|
|
|
||
|
|
tol : float, default=1e-6
|
||
|
|
Tolerance for 'arpack' method
|
||
|
|
Not used if eigen_solver=='dense'.
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
maximum number of iterations for the arpack solver.
|
||
|
|
Not used if eigen_solver=='dense'.
|
||
|
|
|
||
|
|
method : {'standard', 'hessian', 'modified', 'ltsa'}, default='standard'
|
||
|
|
- `standard`: use the standard locally linear embedding algorithm. see
|
||
|
|
reference [1]_
|
||
|
|
- `hessian`: use the Hessian eigenmap method. This method requires
|
||
|
|
``n_neighbors > n_components * (1 + (n_components + 1) / 2``. see
|
||
|
|
reference [2]_
|
||
|
|
- `modified`: use the modified locally linear embedding algorithm.
|
||
|
|
see reference [3]_
|
||
|
|
- `ltsa`: use local tangent space alignment algorithm. see
|
||
|
|
reference [4]_
|
||
|
|
|
||
|
|
hessian_tol : float, default=1e-4
|
||
|
|
Tolerance for Hessian eigenmapping method.
|
||
|
|
Only used if ``method == 'hessian'``
|
||
|
|
|
||
|
|
modified_tol : float, default=1e-12
|
||
|
|
Tolerance for modified LLE method.
|
||
|
|
Only used if ``method == 'modified'``
|
||
|
|
|
||
|
|
neighbors_algorithm : {'auto', 'brute', 'kd_tree', 'ball_tree'}, \
|
||
|
|
default='auto'
|
||
|
|
algorithm to use for nearest neighbors search,
|
||
|
|
passed to neighbors.NearestNeighbors instance
|
||
|
|
|
||
|
|
random_state : int, RandomState instance, default=None
|
||
|
|
Determines the random number generator when
|
||
|
|
``eigen_solver`` == 'arpack'. Pass an int for reproducible results
|
||
|
|
across multiple function calls. See :term: `Glossary <random_state>`.
|
||
|
|
|
||
|
|
n_jobs : int or None, default=None
|
||
|
|
The number of parallel jobs to run.
|
||
|
|
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
|
||
|
|
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
|
||
|
|
for more details.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
embedding_ : array-like, shape [n_samples, n_components]
|
||
|
|
Stores the embedding vectors
|
||
|
|
|
||
|
|
reconstruction_error_ : float
|
||
|
|
Reconstruction error associated with `embedding_`
|
||
|
|
|
||
|
|
nbrs_ : NearestNeighbors object
|
||
|
|
Stores nearest neighbors instance, including BallTree or KDtree
|
||
|
|
if applicable.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from sklearn.datasets import load_digits
|
||
|
|
>>> from sklearn.manifold import LocallyLinearEmbedding
|
||
|
|
>>> X, _ = load_digits(return_X_y=True)
|
||
|
|
>>> X.shape
|
||
|
|
(1797, 64)
|
||
|
|
>>> embedding = LocallyLinearEmbedding(n_components=2)
|
||
|
|
>>> X_transformed = embedding.fit_transform(X[:100])
|
||
|
|
>>> X_transformed.shape
|
||
|
|
(100, 2)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] Roweis, S. & Saul, L. Nonlinear dimensionality reduction
|
||
|
|
by locally linear embedding. Science 290:2323 (2000).
|
||
|
|
.. [2] Donoho, D. & Grimes, C. Hessian eigenmaps: Locally
|
||
|
|
linear embedding techniques for high-dimensional data.
|
||
|
|
Proc Natl Acad Sci U S A. 100:5591 (2003).
|
||
|
|
.. [3] Zhang, Z. & Wang, J. MLLE: Modified Locally Linear
|
||
|
|
Embedding Using Multiple Weights.
|
||
|
|
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.382
|
||
|
|
.. [4] Zhang, Z. & Zha, H. Principal manifolds and nonlinear
|
||
|
|
dimensionality reduction via tangent space alignment.
|
||
|
|
Journal of Shanghai Univ. 8:406 (2004)
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def __init__(self, *, n_neighbors=5, n_components=2, reg=1E-3,
|
||
|
|
eigen_solver='auto', tol=1E-6, max_iter=100,
|
||
|
|
method='standard', hessian_tol=1E-4, modified_tol=1E-12,
|
||
|
|
neighbors_algorithm='auto', random_state=None, n_jobs=None):
|
||
|
|
self.n_neighbors = n_neighbors
|
||
|
|
self.n_components = n_components
|
||
|
|
self.reg = reg
|
||
|
|
self.eigen_solver = eigen_solver
|
||
|
|
self.tol = tol
|
||
|
|
self.max_iter = max_iter
|
||
|
|
self.method = method
|
||
|
|
self.hessian_tol = hessian_tol
|
||
|
|
self.modified_tol = modified_tol
|
||
|
|
self.random_state = random_state
|
||
|
|
self.neighbors_algorithm = neighbors_algorithm
|
||
|
|
self.n_jobs = n_jobs
|
||
|
|
|
||
|
|
def _fit_transform(self, X):
|
||
|
|
self.nbrs_ = NearestNeighbors(n_neighbors=self.n_neighbors,
|
||
|
|
algorithm=self.neighbors_algorithm,
|
||
|
|
n_jobs=self.n_jobs)
|
||
|
|
|
||
|
|
random_state = check_random_state(self.random_state)
|
||
|
|
X = self._validate_data(X, dtype=float)
|
||
|
|
self.nbrs_.fit(X)
|
||
|
|
self.embedding_, self.reconstruction_error_ = \
|
||
|
|
locally_linear_embedding(
|
||
|
|
X=self.nbrs_, n_neighbors=self.n_neighbors,
|
||
|
|
n_components=self.n_components,
|
||
|
|
eigen_solver=self.eigen_solver, tol=self.tol,
|
||
|
|
max_iter=self.max_iter, method=self.method,
|
||
|
|
hessian_tol=self.hessian_tol, modified_tol=self.modified_tol,
|
||
|
|
random_state=random_state, reg=self.reg, n_jobs=self.n_jobs)
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Compute the embedding vectors for data X
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape [n_samples, n_features]
|
||
|
|
training set.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self : returns an instance of self.
|
||
|
|
"""
|
||
|
|
self._fit_transform(X)
|
||
|
|
return self
|
||
|
|
|
||
|
|
def fit_transform(self, X, y=None):
|
||
|
|
"""Compute the embedding vectors for data X and transform X.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape [n_samples, n_features]
|
||
|
|
training set.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
X_new : array-like, shape (n_samples, n_components)
|
||
|
|
"""
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||
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|
self._fit_transform(X)
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||
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|
return self.embedding_
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||
|
|
|
||
|
|
def transform(self, X):
|
||
|
|
"""
|
||
|
|
Transform new points into embedding space.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape (n_samples, n_features)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
X_new : array, shape = [n_samples, n_components]
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Because of scaling performed by this method, it is discouraged to use
|
||
|
|
it together with methods that are not scale-invariant (like SVMs)
|
||
|
|
"""
|
||
|
|
check_is_fitted(self)
|
||
|
|
|
||
|
|
X = check_array(X)
|
||
|
|
ind = self.nbrs_.kneighbors(X, n_neighbors=self.n_neighbors,
|
||
|
|
return_distance=False)
|
||
|
|
weights = barycenter_weights(X, self.nbrs_._fit_X, ind, reg=self.reg)
|
||
|
|
X_new = np.empty((X.shape[0], self.n_components))
|
||
|
|
for i in range(X.shape[0]):
|
||
|
|
X_new[i] = np.dot(self.embedding_[ind[i]].T, weights[i])
|
||
|
|
return X_new
|