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"""Locally Linear Embedding"""
# Author: Fabian Pedregosa -- <fabian.pedregosa@inria.fr>
# Jake Vanderplas -- <vanderplas@astro.washington.edu>
# License: BSD 3 clause (C) INRIA 2011
import numpy as np
from scipy.linalg import eigh, svd, qr, solve
from scipy.sparse import eye, csr_matrix
from scipy.sparse.linalg import eigsh
from ..base import BaseEstimator, TransformerMixin, _UnstableArchMixin
from ..utils import check_random_state, check_array
from ..utils.extmath import stable_cumsum
from ..utils.validation import check_is_fitted
from ..utils.validation import FLOAT_DTYPES
from ..utils.validation import _deprecate_positional_args
from ..neighbors import NearestNeighbors
def barycenter_weights(X, Z, reg=1e-3):
"""Compute barycenter weights of X from Y along the first axis
We estimate the weights to assign to each point in Y[i] to recover
the point X[i]. The barycenter weights sum to 1.
Parameters
----------
X : array-like, shape (n_samples, n_dim)
Z : array-like, shape (n_samples, n_neighbors, n_dim)
reg : float, optional
amount of regularization to add for the problem to be
well-posed in the case of n_neighbors > n_dim
Returns
-------
B : array-like, shape (n_samples, n_neighbors)
Notes
-----
See developers note for more information.
"""
X = check_array(X, dtype=FLOAT_DTYPES)
Z = check_array(Z, dtype=FLOAT_DTYPES, allow_nd=True)
n_samples, n_neighbors = X.shape[0], Z.shape[1]
B = np.empty((n_samples, n_neighbors), dtype=X.dtype)
v = np.ones(n_neighbors, dtype=X.dtype)
# this might raise a LinalgError if G is singular and has trace
# zero
for i, A in enumerate(Z.transpose(0, 2, 1)):
C = A.T - X[i] # broadcasting
G = np.dot(C, C.T)
trace = np.trace(G)
if trace > 0:
R = reg * trace
else:
R = reg
G.flat[::Z.shape[1] + 1] += R
w = solve(G, v, sym_pos=True)
B[i, :] = w / np.sum(w)
return B
def barycenter_kneighbors_graph(X, n_neighbors, reg=1e-3, n_jobs=None):
"""Computes the barycenter weighted graph of k-Neighbors for points in X
Parameters
----------
X : {array-like, NearestNeighbors}
Sample data, shape = (n_samples, n_features), in the form of a
numpy array or a NearestNeighbors object.
n_neighbors : int
Number of neighbors for each sample.
reg : float, optional
Amount of regularization when solving the least-squares
problem. Only relevant if mode='barycenter'. If None, use the
default.
n_jobs : int or None, optional (default=None)
The number of parallel jobs to run for neighbors search.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
Returns
-------
A : sparse matrix in CSR format, shape = [n_samples, n_samples]
A[i, j] is assigned the weight of edge that connects i to j.
See also
--------
sklearn.neighbors.kneighbors_graph
sklearn.neighbors.radius_neighbors_graph
"""
knn = NearestNeighbors(n_neighbors=n_neighbors + 1, n_jobs=n_jobs).fit(X)
X = knn._fit_X
n_samples = knn.n_samples_fit_
ind = knn.kneighbors(X, return_distance=False)[:, 1:]
data = barycenter_weights(X, X[ind], reg=reg)
indptr = np.arange(0, n_samples * n_neighbors + 1, n_neighbors)
return csr_matrix((data.ravel(), ind.ravel(), indptr),
shape=(n_samples, n_samples))
def null_space(M, k, k_skip=1, eigen_solver='arpack', tol=1E-6, max_iter=100,
random_state=None):
"""
Find the null space of a matrix M.
Parameters
----------
M : {array, matrix, sparse matrix, LinearOperator}
Input covariance matrix: should be symmetric positive semi-definite
k : integer
Number of eigenvalues/vectors to return
k_skip : integer, optional
Number of low eigenvalues to skip.
eigen_solver : string, {'auto', 'arpack', 'dense'}
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, optional
Tolerance for 'arpack' method.
Not used if eigen_solver=='dense'.
max_iter : int
Maximum number of iterations for 'arpack' method.
Not used if eigen_solver=='dense'
random_state : int, RandomState instance, default=None
Determines the random number generator when ``solver`` == 'arpack'.
Pass an int for reproducible results across multiple function calls.
See :term: `Glossary <random_state>`.
"""
if eigen_solver == 'auto':
if M.shape[0] > 200 and k + k_skip < 10:
eigen_solver = 'arpack'
else:
eigen_solver = 'dense'
if eigen_solver == 'arpack':
random_state = check_random_state(random_state)
# initialize with [-1,1] as in ARPACK
v0 = random_state.uniform(-1, 1, M.shape[0])
try:
eigen_values, eigen_vectors = eigsh(M, k + k_skip, sigma=0.0,
tol=tol, maxiter=max_iter,
v0=v0)
except RuntimeError as msg:
raise ValueError("Error in determining null-space with ARPACK. "
"Error message: '%s'. "
"Note that method='arpack' can fail when the "
"weight matrix is singular or otherwise "
"ill-behaved. method='dense' is recommended. "
"See online documentation for more information."
% msg)
return eigen_vectors[:, k_skip:], np.sum(eigen_values[k_skip:])
elif eigen_solver == 'dense':
if hasattr(M, 'toarray'):
M = M.toarray()
eigen_values, eigen_vectors = eigh(
M, eigvals=(k_skip, k + k_skip - 1), overwrite_a=True)
index = np.argsort(np.abs(eigen_values))
return eigen_vectors[:, index], np.sum(eigen_values)
else:
raise ValueError("Unrecognized eigen_solver '%s'" % eigen_solver)
@_deprecate_positional_args
def locally_linear_embedding(
X, *, n_neighbors, n_components, reg=1e-3, eigen_solver='auto',
tol=1e-6, max_iter=100, method='standard', hessian_tol=1E-4,
modified_tol=1E-12, random_state=None, n_jobs=None):
"""Perform a Locally Linear Embedding analysis on the data.
Read more in the :ref:`User Guide <locally_linear_embedding>`.
Parameters
----------
X : {array-like, NearestNeighbors}
Sample data, shape = (n_samples, n_features), in the form of a
numpy array or a NearestNeighbors object.
n_neighbors : integer
number of neighbors to consider for each point.
n_components : integer
number of coordinates for the manifold.
reg : float
regularization constant, multiplies the trace of the local covariance
matrix of the distances.
eigen_solver : string, {'auto', 'arpack', 'dense'}
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, optional
Tolerance for 'arpack' method
Not used if eigen_solver=='dense'.
max_iter : integer
maximum number of iterations for the arpack solver.
method : {'standard', 'hessian', 'modified', 'ltsa'}
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, optional
Tolerance for Hessian eigenmapping method.
Only used if method == 'hessian'
modified_tol : float, optional
Tolerance for modified LLE method.
Only used if method == 'modified'
random_state : int, RandomState instance, default=None
Determines the random number generator when ``solver`` == 'arpack'.
Pass an int for reproducible results across multiple function calls.
See :term: `Glossary <random_state>`.
n_jobs : int or None, optional (default=None)
The number of parallel jobs to run for neighbors search.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
Returns
-------
Y : array-like, shape [n_samples, n_components]
Embedding vectors.
squared_error : float
Reconstruction error for the embedding vectors. Equivalent to
``norm(Y - W Y, 'fro')**2``, where W are the reconstruction weights.
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)
"""
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
if method not in ('standard', 'hessian', 'modified', 'ltsa'):
raise ValueError("unrecognized method '%s'" % method)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1, n_jobs=n_jobs)
nbrs.fit(X)
X = nbrs._fit_X
N, d_in = X.shape
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if n_neighbors >= N:
raise ValueError(
"Expected n_neighbors <= n_samples, "
" but n_samples = %d, n_neighbors = %d" %
(N, n_neighbors)
)
if n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
M_sparse = (eigen_solver != 'dense')
if method == 'standard':
W = barycenter_kneighbors_graph(
nbrs, n_neighbors=n_neighbors, reg=reg, n_jobs=n_jobs)
# we'll compute M = (I-W)'(I-W)
# depending on the solver, we'll do this differently
if M_sparse:
M = eye(*W.shape, format=W.format) - W
M = (M.T * M).tocsr()
else:
M = (W.T * W - W.T - W).toarray()
M.flat[::M.shape[0] + 1] += 1 # W = W - I = W - I
elif method == 'hessian':
dp = n_components * (n_components + 1) // 2
if n_neighbors <= n_components + dp:
raise ValueError("for method='hessian', n_neighbors must be "
"greater than "
"[n_components * (n_components + 3) / 2]")
neighbors = nbrs.kneighbors(X, n_neighbors=n_neighbors + 1,
return_distance=False)
neighbors = neighbors[:, 1:]
Yi = np.empty((n_neighbors, 1 + n_components + dp), dtype=np.float64)
Yi[:, 0] = 1
M = np.zeros((N, N), dtype=np.float64)
use_svd = (n_neighbors > d_in)
for i in range(N):
Gi = X[neighbors[i]]
Gi -= Gi.mean(0)
# build Hessian estimator
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 : integer
number of neighbors to consider for each point.
n_components : integer
number of coordinates for the manifold
reg : float
regularization constant, multiplies the trace of the local covariance
matrix of the distances.
eigen_solver : string, {'auto', 'arpack', 'dense'}
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, optional
Tolerance for 'arpack' method
Not used if eigen_solver=='dense'.
max_iter : integer
maximum number of iterations for the arpack solver.
Not used if eigen_solver=='dense'.
method : string ('standard', 'hessian', 'modified' or 'ltsa')
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, optional
Tolerance for Hessian eigenmapping method.
Only used if ``method == 'hessian'``
modified_tol : float, optional
Tolerance for modified LLE method.
Only used if ``method == 'modified'``
neighbors_algorithm : string ['auto'|'brute'|'kd_tree'|'ball_tree']
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, 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
----------
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)
"""
self._fit_transform(X)
return self.embedding_
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