scikit-learn · GaelVaroquaux · Oct 23, 2015 · Jul 21, 2015 · Jul 24, 2015 · Jul 31, 2015
diff --git a/doc/modules/pipeline.rst b/doc/modules/pipeline.rst
@@ -156,7 +156,7 @@ and ``value`` is an estimator object::
         n_components=None, whiten=False)), ('kernel_pca', KernelPCA(alpha=1.0,
         coef0=1, degree=3, eigen_solver='auto', fit_inverse_transform=False,
         gamma=None, kernel='linear', kernel_params=None, max_iter=None,
-        n_components=None, remove_zero_eig=False, tol=0))],
+        n_components=None, random_state=None, remove_zero_eig=False, tol=0))],
         transformer_weights=None)
 
 Like pipelines, feature unions have a shorthand constructor called

diff --git a/doc/whats_new.rst b/doc/whats_new.rst
@@ -43,6 +43,10 @@ Bug fixes
     - Fixed bug in :func:`manifold.spectral_embedding` where diagonal of unnormalized
       Laplacian matrix was incorrectly set to 1. By `Peter Fischer`_.
 
+    - Fixed incorrect initialization of :func:`utils.arpack.eigsh` on all 
+      occurrences. Affects :class:`cluster.SpectralBiclustering`,
+      :class:`decomposition.KernelPCA`, :class:`manifold.LocallyLinearEmbedding`,
+      and :class:`manifold.SpectralEmbedding`. By `Peter Fischer`_.
 
 API changes summary
 -------------------

diff --git a/sklearn/cluster/bicluster.py b/sklearn/cluster/bicluster.py
@@ -14,6 +14,7 @@
 from . import KMeans, MiniBatchKMeans
 from ..base import BaseEstimator, BiclusterMixin
 from ..externals import six
+from ..utils import check_random_state
 from ..utils.arpack import eigsh, svds
 
 from ..utils.extmath import (make_nonnegative, norm, randomized_svd,
@@ -140,12 +141,18 @@ def _svd(self, array, n_components, n_discard):
                 # some eigenvalues of A * A.T are negative, causing
                 # sqrt() to be np.nan. This causes some vectors in vt
                 # to be np.nan.
-                _, v = eigsh(safe_sparse_dot(array.T, array),
-                             ncv=self.n_svd_vecs)
+                A = safe_sparse_dot(array.T, array)
+                random_state = check_random_state(self.random_state)
+                # initialize with [-1,1] as in ARPACK
+                v0 = random_state.uniform(-1, 1, A.shape[0])
+                _, v = eigsh(A, ncv=self.n_svd_vecs, v0=v0)
                 vt = v.T
             if np.any(np.isnan(u)):
-                _, u = eigsh(safe_sparse_dot(array, array.T),
-                             ncv=self.n_svd_vecs)
+                A = safe_sparse_dot(array, array.T)
+                random_state = check_random_state(self.random_state)
+                # initialize with [-1,1] as in ARPACK
+                v0 = random_state.uniform(-1, 1, A.shape[0])
+                _, u = eigsh(A, ncv=self.n_svd_vecs, v0=v0)
 
         assert_all_finite(u)
         assert_all_finite(vt)

diff --git a/sklearn/decomposition/kernel_pca.py b/sklearn/decomposition/kernel_pca.py
@@ -6,6 +6,7 @@
 import numpy as np
 from scipy import linalg
 
+from ..utils import check_random_state
 from ..utils.arpack import eigsh
 from ..utils.validation import check_is_fitted
 from ..exceptions import NotFittedError
@@ -76,6 +77,10 @@ class KernelPCA(BaseEstimator, TransformerMixin):
         When n_components is None, this parameter is ignored and components
         with zero eigenvalues are removed regardless.
 
+    random_state : int seed, RandomState instance, or None, default : None
+        A pseudo random number generator used for the initialization of the
+        residuals when eigen_solver == 'arpack'.
+
     Attributes
     ----------
 
@@ -103,7 +108,8 @@ class KernelPCA(BaseEstimator, TransformerMixin):
     def __init__(self, n_components=None, kernel="linear",
                  gamma=None, degree=3, coef0=1, kernel_params=None,
                  alpha=1.0, fit_inverse_transform=False, eigen_solver='auto',
-                 tol=0, max_iter=None, remove_zero_eig=False):
+                 tol=0, max_iter=None, remove_zero_eig=False,
+                 random_state=None):
         if fit_inverse_transform and kernel == 'precomputed':
             raise ValueError(
                 "Cannot fit_inverse_transform with a precomputed kernel.")
@@ -120,6 +126,7 @@ def __init__(self, n_components=None, kernel="linear",
         self.tol = tol
         self.max_iter = max_iter
         self._centerer = KernelCenterer()
+        self.random_state = random_state
 
     @property
     def _pairwise(self):
@@ -158,10 +165,14 @@ def _fit_transform(self, K):
             self.lambdas_, self.alphas_ = linalg.eigh(
                 K, eigvals=(K.shape[0] - n_components, K.shape[0] - 1))
         elif eigen_solver == 'arpack':
+            random_state = check_random_state(self.random_state)
+            # initialize with [-1,1] as in ARPACK
+            v0 = random_state.uniform(-1, 1, K.shape[0])
             self.lambdas_, self.alphas_ = eigsh(K, n_components,
                                                 which="LA",
                                                 tol=self.tol,
-                                                maxiter=self.max_iter)
+                                                maxiter=self.max_iter,
+                                                v0=v0)
 
         # sort eigenvectors in descending order
         indices = self.lambdas_.argsort()[::-1]

diff --git a/sklearn/manifold/locally_linear.py b/sklearn/manifold/locally_linear.py
@@ -148,7 +148,8 @@ def null_space(M, k, k_skip=1, eigen_solver='arpack', tol=1E-6, max_iter=100,
 
     if eigen_solver == 'arpack':
         random_state = check_random_state(random_state)
-        v0 = random_state.rand(M.shape[0])
+        # 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,

diff --git a/sklearn/manifold/spectral_embedding_.py b/sklearn/manifold/spectral_embedding_.py
@@ -259,9 +259,10 @@ def spectral_embedding(adjacency, n_components=8, eigen_solver=None,
             # We are computing the opposite of the laplacian inplace so as
             # to spare a memory allocation of a possibly very large array
             laplacian *= -1
+            v0 = random_state.uniform(-1, 1, laplacian.shape[0])
             lambdas, diffusion_map = eigsh(laplacian, k=n_components,
                                            sigma=1.0, which='LM',
-                                           tol=eigen_tol)
+                                           tol=eigen_tol, v0=v0)
             embedding = diffusion_map.T[n_components::-1] * dd
         except RuntimeError:
             # When submatrices are exactly singular, an LU decomposition

diff --git a/sklearn/utils/tests/test_utils.py b/sklearn/utils/tests/test_utils.py
@@ -3,11 +3,13 @@
 import numpy as np
 import scipy.sparse as sp
 from scipy.linalg import pinv2
+from scipy.linalg import eigh
 from itertools import chain
 
 from sklearn.utils.testing import (assert_equal, assert_raises, assert_true,
                                    assert_almost_equal, assert_array_equal,
-                                   SkipTest, assert_raises_regex)
+                                   SkipTest, assert_raises_regex,
+                                   assert_greater_equal)
 
 from sklearn.utils import check_random_state
 from sklearn.utils import deprecated
@@ -18,7 +20,9 @@
 from sklearn.utils import shuffle
 from sklearn.utils import gen_even_slices
 from sklearn.utils.extmath import pinvh
+from sklearn.utils.arpack import eigsh
 from sklearn.utils.mocking import MockDataFrame
+from sklearn.utils.graph import graph_laplacian
 
 
 def test_make_rng():
@@ -126,6 +130,26 @@ def test_pinvh_simple_complex():
     assert_almost_equal(np.dot(a, a_pinv), np.eye(3))
 
 
+def test_arpack_eigsh_initialization():
+    # Non-regression test that shows null-space computation is better with 
+    # initialization of eigsh from [-1,1] instead of [0,1]
+    random_state = check_random_state(42)
+
+    A = random_state.rand(50, 50)
+    A = np.dot(A.T, A)  # create s.p.d. matrix
+    A = graph_laplacian(A) + 1e-7 * np.identity(A.shape[0])
+    k = 5
+
+    # Test if eigsh is working correctly
+    # New initialization [-1,1] (as in original ARPACK)
+    # Was [0,1] before, with which this test could fail
+    v0 = random_state.uniform(-1,1, A.shape[0])
+    w, _ = eigsh(A, k=k, sigma=0.0, v0=v0)
+
+    # Eigenvalues of s.p.d. matrix should be nonnegative, w[0] is smallest
+    assert_greater_equal(w[0], 0)
+
+
 def test_column_or_1d():
     EXAMPLES = [
         ("binary", ["spam", "egg", "spam"]),