def test_connectivity(seed=36):
# Test that graph connectivity test works as expected
graph = np.array([[1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 1, 1, 1, 0], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert_equal(_graph_is_connected(graph), False)
assert_equal(_graph_is_connected(csr_matrix(graph)), False)
assert_equal(_graph_is_connected(csc_matrix(graph)), False)
graph = np.array([[1, 1, 0, 0, 0], [1, 1, 1, 0, 0], [0, 1, 1, 1, 0], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert_equal(_graph_is_connected(graph), True)
assert_equal(_graph_is_connected(csr_matrix(graph)), True)
assert_equal(_graph_is_connected(csc_matrix(graph)), True)
def compute_markov_matrix(self, skip_checks=False, overwrite=False):
"""
Slightly modified code originally written by Satrajit Ghosh ([email protected] github.com/satra/mapalign)
"""
import numpy as np
import scipy.sparse as sps
L = self.current_cmat_
alpha = self.diff_alpha
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not skip_checks:
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
if not _graph_is_connected(L):
raise ValueError('Graph is disconnected')
ndim = L.shape[0]
if overwrite:
L_alpha = L
else:
L_alpha = L.copy()
if alpha > 0:
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
L_alpha = L_alpha * d_alpha[np.newaxis, :]
# Step 3
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
return L_alpha
def compute_markov_matrix(L,
alpha=0.5,
diffusion_time=0,
skip_checks=False,
overwrite=False):
"""
Computes a markov transition matrix from the affinity matrix.
Code written by Satra Ghosh (https://github.com/satra/mapalign)
"""
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not skip_checks:
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
if not _graph_is_connected(L):
raise ValueError('Graph is disconnected')
ndim = L.shape[0]
if overwrite:
L_alpha = L
else:
L_alpha = L.copy()
if alpha > 0:
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
L_alpha = L_alpha * d_alpha[np.newaxis, :]
# Step 3
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
return L_alpha
def test_connectivity(seed=36):
# Test that graph connectivity test works as expected
graph = np.array([[1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 1, 1, 1, 0],
[0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert not _graph_is_connected(graph)
assert not _graph_is_connected(sparse.csr_matrix(graph))
assert not _graph_is_connected(sparse.csc_matrix(graph))
graph = np.array([[1, 1, 0, 0, 0], [1, 1, 1, 0, 0], [0, 1, 1, 1, 0],
[0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert _graph_is_connected(graph)
assert _graph_is_connected(sparse.csr_matrix(graph))
assert _graph_is_connected(sparse.csc_matrix(graph))
def test_connectivity(seed=36):
# Test that graph connectivity test works as expected
graph = np.array([[1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 1, 1, 1, 0],
[0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert_equal(_graph_is_connected(graph), False)
assert_equal(_graph_is_connected(csr_matrix(graph)), False)
assert_equal(_graph_is_connected(csc_matrix(graph)), False)
graph = np.array([[1, 1, 0, 0, 0], [1, 1, 1, 0, 0], [0, 1, 1, 1, 0],
[0, 0, 1, 1, 1], [0, 0, 0, 1, 1]])
assert_equal(_graph_is_connected(graph), True)
assert_equal(_graph_is_connected(csr_matrix(graph)), True)
assert_equal(_graph_is_connected(csc_matrix(graph)), True)
def compute_markov_matrix(L, alpha=0.5, diffusion_time=0, skip_checks=False, overwrite=False):
import numpy as np
import scipy.sparse as sps
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not skip_checks:
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
if not _graph_is_connected(L):
raise ValueError('Graph is disconnected')
ndim = L.shape[0]
if overwrite:
L_alpha = L
else:
L_alpha = L.copy()
if alpha > 0:
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
L_alpha = L_alpha * d_alpha[np.newaxis, :]
# Step 3
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
return L_alpha
def diffusemapReduction(dist,
epi=0.5,
mode='manual',
n_neighbors=10,
n_components=10):
if mode == 'auto':
dist_thresh, epi = thresholdDistanceMatrix(dist)
else:
dist_thresh, epi = thresholdDistanceMatrix(dist,
epi=epi,
mode='manual')
plt.figure(10)
plt.imshow(dist_thresh)
plt.colorbar()
if not _graph_is_connected(dist_thresh):
raise ValueError('Graph is disconnected')
embedding, result = compute_diffusion_map(dist_thresh,
alpha=0.,
n_components=n_components)
result['epi'] = epi
return embedding, result
def compute_diffusion_map(L,
alpha=0.5,
n_components=None,
diffusion_time=0,
skip_checks=False,
overwrite=False):
"""Compute the diffusion maps of a symmetric similarity matrix
L : matrix N x N
L is symmetric and L(x, y) >= 0
alpha: float [0, 1]
Setting alpha=1 and the diffusion operator approximates the
Laplace-Beltrami operator. We then recover the Riemannian geometry
of the data set regardless of the distribution of the points. To
describe the long-term behavior of the point distribution of a
system of stochastic differential equations, we can use alpha=0.5
and the resulting Markov chain approximates the Fokker-Planck
diffusion. With alpha=0, it reduces to the classical graph Laplacian
normalization.
n_components: int
The number of diffusion map components to return. Due to the
spectrum decay of the eigenvalues, only a few terms are necessary to
achieve a given relative accuracy in the sum M^t.
diffusion_time: float >= 0
use the diffusion_time (t) step transition matrix M^t
t not only serves as a time parameter, but also has the dual role of
scale parameter. One of the main ideas of diffusion framework is
that running the chain forward in time (taking larger and larger
powers of M) reveals the geometric structure of X at larger and
larger scales (the diffusion process).
t = 0 empirically provides a reasonable balance from a clustering
perspective. Specifically, the notion of a cluster in the data set
is quantified as a region in which the probability of escaping this
region is low (within a certain time t).
skip_checks: bool
Avoid expensive pre-checks on input data. The caller has to make
sure that input data is valid or results will be undefined.
overwrite: bool
Optimize memory usage by re-using input matrix L as scratch space.
References
----------
[1] https://en.wikipedia.org/wiki/Diffusion_map
[2] Coifman, R.R.; S. Lafon. (2006). "Diffusion maps". Applied and
Computational Harmonic Analysis 21: 5-30. doi:10.1016/j.acha.2006.04.006
"""
import numpy as np
import scipy.sparse as sps
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not skip_checks:
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
if not _graph_is_connected(L):
raise ValueError('Graph is disconnected')
ndim = L.shape[0]
if overwrite:
L_alpha = L
else:
L_alpha = L.copy()
if alpha > 0:
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
L_alpha = L_alpha * d_alpha[np.newaxis, :]
# Step 3
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
M = L_alpha
from scipy.sparse.linalg import eigsh, eigs
# Step 4
func = eigs
if n_components is not None:
lambdas, vectors = func(M, k=n_components + 1)
else:
lambdas, vectors = func(M, k=max(2, int(np.sqrt(ndim))))
del M
if func == eigsh:
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
else:
lambdas = np.real(lambdas)
vectors = np.real(vectors)
lambda_idx = np.argsort(lambdas)[::-1]
lambdas = lambdas[lambda_idx]
vectors = vectors[:, lambda_idx]
# Step 5
psi = vectors / vectors[:, [0]]
diffusion_times = diffusion_time
if diffusion_time == 0:
diffusion_times = np.exp(1. -
np.log(1 - lambdas[1:]) / np.log(lambdas[1:]))
lambdas = lambdas[1:] / (1 - lambdas[1:])
else:
lambdas = lambdas[1:]**float(diffusion_time)
lambda_ratio = lambdas / lambdas[0]
threshold = max(0.05, lambda_ratio[-1])
n_components_auto = np.amax(np.nonzero(lambda_ratio > threshold)[0])
n_components_auto = min(n_components_auto, ndim)
if n_components is None:
n_components = n_components_auto
embedding = psi[:, 1:(n_components + 1)] * lambdas[:n_components][None, :]
result = dict(lambdas=lambdas,
vectors=vectors,
n_components=n_components,
diffusion_time=diffusion_times,
n_components_auto=n_components_auto)
return embedding, result
def diffusion_mapping(adj,
n_components=10,
alpha=0.5,
diffusion_time=0,
random_state=None):
"""Compute diffusion map of affinity matrix.
Parameters
----------
adj : ndarray or sparse matrix, shape = (n, n)
Affinity matrix.
n_components : int or None, optional
Number of eigenvectors. If None, selection of `n_components` is based
on 95% drop-off in eigenvalues. When `n_components` is None,
the maximum number of eigenvectors is restricted to
``n_components <= sqrt(n)``. Default is 10.
alpha : float, optional
Anisotropic diffusion parameter, ``0 <= alpha <= 1``. Default is 0.5.
diffusion_time : int, optional
Diffusion time or scale. If ``diffusion_time == 0`` use multi-scale
diffusion maps. Default is 0.
random_state : int or None, optional
Random state. Default is None.
Returns
-------
v : ndarray, shape (n, n_components)
Eigenvectors of the affinity matrix in same order.
w : ndarray, shape (n_components,)
Eigenvalues of the affinity matrix in descending order.
References
----------
* Coifman, R.R.; S. Lafon. (2006). "Diffusion maps". Applied and
Computational Harmonic Analysis 21: 5-30. doi:10.1016/j.acha.2006.04.006
* Joseph W.R., Peter E.F., Ann B.L., Chad M.S. Accurate parameter
estimation for star formation history in galaxies using SDSS spectra.
"""
rs = check_random_state(random_state)
use_sparse = ssp.issparse(adj)
# Make symmetric
if not is_symmetric(adj, tol=1E-10):
warnings.warn('Affinity is not symmetric. Making symmetric.')
adj = make_symmetric(adj, check=False, copy=True, sparse_format='coo')
else: # Copy anyways because we will be working on the matrix
adj = adj.tocoo(copy=True) if use_sparse else adj.copy()
# Check connected
if not _graph_is_connected(adj):
warnings.warn('Graph is not fully connected.')
###########################################################
# Step 2
###########################################################
# When α=0, you get back the diffusion map based on the random walk-style
# diffusion operator (and Laplacian Eigenmaps). For α=1, the diffusion
# operator approximates the Laplace-Beltrami operator and for α=0.5,
# you get Fokker-Planck diffusion. The anisotropic diffusion
# parameter: \alpha \in \[0, 1\]
# W(α) = D^{−1/\alpha} W D^{−1/\alpha}
if alpha > 0:
if use_sparse:
d = np.power(adj.sum(axis=1).A1, -alpha)
adj.data *= d[adj.row]
adj.data *= d[adj.col]
else:
d = adj.sum(axis=1, keepdims=True)
d = np.power(d, -alpha)
adj *= d.T
adj *= d
###########################################################
# Step 3
###########################################################
# Diffusion operator
# P(α) = D(α)^{−1}W(α)
if use_sparse:
d_alpha = np.power(adj.sum(axis=1).A1, -1)
adj.data *= d_alpha[adj.row]
else:
adj *= np.power(adj.sum(axis=1, keepdims=True), -1)
###########################################################
# Step 4
###########################################################
if n_components is None:
n_components = max(2, int(np.sqrt(adj.shape[0])))
auto_n_comp = True
else:
auto_n_comp = False
# For repeatability of results
v0 = rs.uniform(-1, 1, adj.shape[0])
# Find largest eigenvalues and eigenvectors
w, v = eigsh(adj, k=n_components + 1, which='LM', tol=0, v0=v0)
# Sort descending
w, v = w[::-1], v[:, ::-1]
###########################################################
# Step 5
###########################################################
# Force first eigenvector to be all ones.
v /= v[:, [0]]
# Largest eigenvalue should be equal to one too
w /= w[0]
# Discard first (largest) eigenvalue and eigenvector
w, v = w[1:], v[:, 1:]
if diffusion_time <= 0:
# use multi-scale diffusion map, ref [4]
# considers all scales: t=1,2,3,...
w /= (1 - w)
else:
# Raise eigenvalues to the power of diffusion time
w **= diffusion_time
if auto_n_comp:
# Choose n_comp to coincide with a 95 % drop-off
# in the eigenvalue multipliers, ref [4]
lambda_ratio = w / w[0]
# If all eigenvalues larger than 0.05, select all
# (i.e., sqrt(adj.shape[0]))
threshold = max(0.05, lambda_ratio[-1])
n_components = np.argmin(lambda_ratio > threshold)
w = w[:n_components]
v = v[:, :n_components]
# Rescale eigenvectors with eigenvalues
v *= w[None, :]
# Consistent sign (s.t. largest value of element eigenvector is pos)
v *= np.sign(v[np.abs(v).argmax(axis=0), range(v.shape[1])])
return v, w
def laplacian_eigenmaps(adj,
n_components=10,
norm_laplacian=True,
random_state=None):
"""Compute embedding using Laplacian eigenmaps.
Adapted from Scikit-learn to also provide eigenvalues.
Parameters
----------
adj : 2D ndarray or sparse matrix
Affinity matrix.
n_components : int, optional
Number of eigenvectors. Default is 10.
norm_laplacian : bool, optional
If True use normalized Laplacian. Default is True.
random_state : int or None, optional
Random state. Default is None.
Returns
-------
v : 2D ndarray, shape (n, n_components)
Eigenvectors of the affinity matrix in same order. Where `n` is
the number of rows of the affinity matrix.
w : 1D ndarray, shape (n_components,)
Eigenvalues of the affinity matrix in ascending order.
References
----------
* Belkin, M. and Niyogi, P. (2003). Laplacian Eigenmaps for
dimensionality reduction and data representation.
Neural Computation 15(6): 1373-96. doi:10.1162/089976603321780317
"""
rs = check_random_state(random_state)
# Make symmetric
if not is_symmetric(adj, tol=1E-10):
warnings.warn('Affinity is not symmetric. Making symmetric.')
adj = make_symmetric(adj, check=False)
# Check connected
if not _graph_is_connected(adj):
warnings.warn('Graph is not fully connected.')
lap, dd = laplacian(adj, normed=norm_laplacian, return_diag=True)
if norm_laplacian:
if ssp.issparse(lap):
lap.setdiag(1)
else:
np.fill_diagonal(lap, 1)
lap *= -1
v0 = rs.uniform(-1, 1, lap.shape[0])
w, v = eigsh(lap, k=n_components + 1, sigma=1, which='LM', tol=0, v0=v0)
# Sort descending and change sign of eigenvalues
w, v = -w[::-1], v[:, ::-1]
if norm_laplacian:
v /= dd[:, None]
# Drop smallest
w, v = w[1:], v[:, 1:]
# Consistent sign (s.t. largest value of element eigenvector is pos)
v *= np.sign(v[np.abs(v).argmax(axis=0), range(v.shape[1])])
return v, w
thresholdDistanceMatrix(dist_Y)
#using diffusion mapping to do this
epi =100
dist_thresh = np.exp(-dist_Y**2/epi/2.0)
M = dist_thresh
for i in range(M.shape[0]):
M[i] = M[i]/np.sum(M[i])
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
plt.figure(10)
plt.imshow(dist_thresh)
plt.colorbar()
if not _graph_is_connected(dist_thresh):
raise ValueError('Graph is disconnected')
embedding, result = compute_diffusion_map(dist_thresh, n_components=5)
# w,v = np.linalg.eig(M)
# U,S,V = np.linalg.svd(M)
# plt.figure(1)
# plt.plot(np.real(v[:,1]))
# plt.figure(2)
# plt.plot(np.real(v[:,2]))
def compute_diffusion_map(L, alpha=0.5, n_components=None, diffusion_time=0, skip_checks=False, overwrite=False):
"""Compute the diffusion maps of a symmetric similarity matrix
L : matrix N x N
L is symmetric and L(x, y) >= 0
alpha: float [0, 1]
Setting alpha=1 and the diffusion operator approximates the
Laplace-Beltrami operator. We then recover the Riemannian geometry
of the data set regardless of the distribution of the points. To
describe the long-term behavior of the point distribution of a
system of stochastic differential equations, we can use alpha=0.5
and the resulting Markov chain approximates the Fokker-Planck
diffusion. With alpha=0, it reduces to the classical graph Laplacian
normalization.
n_components: int
The number of diffusion map components to return. Due to the
spectrum decay of the eigenvalues, only a few terms are necessary to
achieve a given relative accuracy in the sum M^t.
diffusion_time: float >= 0
use the diffusion_time (t) step transition matrix M^t
t not only serves as a time parameter, but also has the dual role of
scale parameter. One of the main ideas of diffusion framework is
that running the chain forward in time (taking larger and larger
powers of M) reveals the geometric structure of X at larger and
larger scales (the diffusion process).
t = 0 empirically provides a reasonable balance from a clustering
perspective. Specifically, the notion of a cluster in the data set
is quantified as a region in which the probability of escaping this
region is low (within a certain time t).
skip_checks: bool
Avoid expensive pre-checks on input data. The caller has to make
sure that input data is valid or results will be undefined.
overwrite: bool
Optimize memory usage by re-using input matrix L as scratch space.
References
----------
[1] https://en.wikipedia.org/wiki/Diffusion_map
[2] Coifman, R.R.; S. Lafon. (2006). "Diffusion maps". Applied and
Computational Harmonic Analysis 21: 5-30. doi:10.1016/j.acha.2006.04.006
"""
import numpy as np
import scipy.sparse as sps
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not skip_checks:
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
if not _graph_is_connected(L):
raise ValueError("Graph is disconnected")
ndim = L.shape[0]
if overwrite:
L_alpha = L
else:
L_alpha = L.copy()
if alpha > 0:
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
L_alpha = L_alpha * d_alpha[np.newaxis, :]
# Step 3
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, np.newaxis] * L_alpha
M = L_alpha
from scipy.sparse.linalg import eigsh, eigs
# Step 4
func = eigs
if n_components is not None:
lambdas, vectors = func(M, k=n_components + 1)
else:
lambdas, vectors = func(M, k=max(2, int(np.sqrt(ndim))))
del M
if func == eigsh:
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
else:
lambdas = np.real(lambdas)
vectors = np.real(vectors)
lambda_idx = np.argsort(lambdas)[::-1]
lambdas = lambdas[lambda_idx]
vectors = vectors[:, lambda_idx]
# Step 5
psi = vectors / vectors[:, [0]]
if diffusion_time == 0:
lambdas = lambdas[1:] / (1 - lambdas[1:])
else:
lambdas = lambdas[1:] ** float(diffusion_time)
lambda_ratio = lambdas / lambdas[0]
threshold = max(0.05, lambda_ratio[-1])
n_components_auto = np.amax(np.nonzero(lambda_ratio > threshold)[0])
n_components_auto = min(n_components_auto, ndim)
if n_components is None:
n_components = n_components_auto
embedding = psi[:, 1 : (n_components + 1)] * lambdas[:n_components][None, :]
result = dict(
lambdas=lambdas,
vectors=vectors,
n_components=n_components,
diffusion_time=diffusion_time,
n_components_auto=n_components_auto,
)
return embedding, result
def compute_diffusion_map(L, alpha=0.5, n_components=None, diffusion_time=0, verbose=False):
# from https://github.com/satra/mapalign/blob/master/mapalign/embed.py
import numpy as np
import scipy.sparse as sps
from sklearn.manifold.spectral_embedding_ import _graph_is_connected
use_sparse = False
if sps.issparse(L):
use_sparse = True
if not _graph_is_connected(L):
raise ValueError('Graph is disconnected')
if verbose:
print 'checked conditions'
ndim = L.shape[0]
L_alpha = L.copy()
if alpha > 0:
if verbose:
print 'step2'
# Step 2
d = np.array(L_alpha.sum(axis=1)).flatten()
d_alpha = np.power(d, -alpha)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
L_alpha.data *= d_alpha[L_alpha.indices]
L_alpha = sps.csr_matrix(L_alpha.transpose().toarray())
else:
L_alpha = d_alpha[:, None] * L_alpha * d_alpha[None, :]
# Step 3
if verbose:
print 'step 3'
d_alpha = np.power(np.array(L_alpha.sum(axis=1)).flatten(), -1)
if use_sparse:
L_alpha.data *= d_alpha[L_alpha.indices]
else:
L_alpha = d_alpha[:, None] * L_alpha
M = L_alpha
from sklearn.utils.arpack import eigsh, eigs
# Step 4
if verbose:
print 'step 4'
func = eigs
if n_components is not None:
lambdas, vectors = func(M, k=n_components + 1)
else:
lambdas, vectors = func(M, k=max(2, int(np.sqrt(ndim))))
del M
if func == eigsh:
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
else:
lambdas = np.real(lambdas)
vectors = np.real(vectors)
lambda_idx = np.argsort(lambdas)[::-1]
lambdas = lambdas[lambda_idx]
vectors = vectors[:, lambda_idx]
# Step 5
if verbose:
print 'step 5'
psi = vectors/vectors[:, [0]]
if diffusion_time == 0:
lambdas = lambdas[1:] / (1 - lambdas[1:])
else:
lambdas = lambdas[1:] ** float(diffusion_time)
lambda_ratio = lambdas/lambdas[0]
threshold = max(0.05, lambda_ratio[-1])
n_components_auto = np.amax(np.nonzero(lambda_ratio > threshold)[0])
n_components_auto = min(n_components_auto, ndim)
if n_components is None:
n_components = n_components_auto
embedding = psi[:, 1:(n_components + 1)] * lambdas[:n_components][None, :]
result = dict(lambdas=lambdas, vectors=vectors,
n_components=n_components, diffusion_time=diffusion_time,
n_components_auto=n_components_auto)
return embedding, result
def spectral_embedding(self,
adjacency,
n_components=8,
eigen_solver=None,
random_state=None,
eigen_tol=0.0,
drop_first=True):
"""
see original at https://github.com/scikit-learn/scikit-learn/blob/14031f6/sklearn/manifold/spectral_embedding_.py#L133
custermize1: return lambdas with the embedded matrix.
custermize2: norm_laplacian is always True
"""
norm_laplacian = True
adjacency = check_symmetric(adjacency)
try:
from pyamg import smoothed_aggregation_solver
except ImportError:
if eigen_solver == "amg":
raise ValueError(
"The eigen_solver was set to 'amg', but pyamg is "
"not available.")
if eigen_solver is None:
eigen_solver = 'arpack'
elif eigen_solver not in ('arpack', 'lobpcg', 'amg'):
raise ValueError("Unknown value for eigen_solver: '%s'."
"Should be 'amg', 'arpack', or 'lobpcg'" %
eigen_solver)
random_state = check_random_state(random_state)
n_nodes = adjacency.shape[0]
# Whether to drop the first eigenvector
if drop_first:
n_components = n_components + 1
if not _graph_is_connected(adjacency):
warnings.warn("Graph is not fully connected, spectral embedding"
" may not work as expected.")
laplacian, dd = graph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
if (eigen_solver == 'arpack' or eigen_solver != 'lobpcg' and
(not sparse.isspmatrix(laplacian) or n_nodes < 5 * n_components)):
# lobpcg used with eigen_solver='amg' has bugs for low number of nodes
# for details see the source code in scipy:
# https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen
# /lobpcg/lobpcg.py#L237
# or matlab:
# http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m
laplacian = _set_diag(laplacian, 1, norm_laplacian)
# Here we'll use shift-invert mode for fast eigenvalues
# (see http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
# for a short explanation of what this means)
# Because the normalized Laplacian has eigenvalues between 0 and 2,
# I - L has eigenvalues between -1 and 1. ARPACK is most efficient
# when finding eigenvalues of largest magnitude (keyword which='LM')
# and when these eigenvalues are very large compared to the rest.
# For very large, very sparse graphs, I - L can have many, many
# eigenvalues very near 1.0. This leads to slow convergence. So
# instead, we'll use ARPACK's shift-invert mode, asking for the
# eigenvalues near 1.0. This effectively spreads-out the spectrum
# near 1.0 and leads to much faster convergence: potentially an
# orders-of-magnitude speedup over simply using keyword which='LA'
# in standard mode.
try:
# We are computing the opposite of the laplacian inplace so as
# to spare a memory allocation of a possibly very large array
laplacian *= -1
lambdas, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which='LM',
tol=eigen_tol)
embedding = diffusion_map.T[n_components::-1] * dd
except RuntimeError:
# When submatrices are exactly singular, an LU decomposition
# in arpack fails. We fallback to lobpcg
eigen_solver = "lobpcg"
# Revert the laplacian to its opposite to have lobpcg work
laplacian *= -1
if eigen_solver == 'amg':
# Use AMG to get a preconditioner and speed up the eigenvalue
# problem.
if not sparse.issparse(laplacian):
warnings.warn("AMG works better for sparse matrices")
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
ml = smoothed_aggregation_solver(check_array(laplacian, 'csr'))
M = ml.aspreconditioner()
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
M=M,
tol=1.e-12,
largest=False)
embedding = diffusion_map.T * dd
if embedding.shape[0] == 1:
raise ValueError
elif eigen_solver == "lobpcg":
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
if n_nodes < 5 * n_components + 1:
# see note above under arpack why lobpcg has problems with small
# number of nodes
# lobpcg will fallback to eigh, so we short circuit it
if sparse.isspmatrix(laplacian):
laplacian = laplacian.toarray()
lambdas, diffusion_map = eigh(laplacian)
embedding = diffusion_map.T[:n_components] * dd
else:
laplacian = _set_diag(laplacian, 1, norm_laplacian)
# We increase the number of eigenvectors requested, as lobpcg
# doesn't behave well in low dimension
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
tol=1e-15,
largest=False,
maxiter=2000)
embedding = diffusion_map.T[:n_components] * dd
if embedding.shape[0] == 1:
raise ValueError
embedding = _deterministic_vector_sign_flip(embedding)
if drop_first:
return embedding[1:n_components].T, lambdas
else:
return embedding[:n_components].T, lambdas