767 lines
24 KiB
Python
767 lines
24 KiB
Python
"""
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Extended math utilities.
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"""
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# Authors: Gael Varoquaux
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# Alexandre Gramfort
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# Alexandre T. Passos
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# Olivier Grisel
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# Lars Buitinck
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# Stefan van der Walt
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# Kyle Kastner
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# Giorgio Patrini
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# License: BSD 3 clause
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from __future__ import division
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import warnings
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import numpy as np
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from scipy import linalg
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from scipy.sparse import issparse, csr_matrix
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from . import check_random_state, deprecated
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from .fixes import np_version
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from .fixes import logsumexp as scipy_logsumexp
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from ._logistic_sigmoid import _log_logistic_sigmoid
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from ..externals.six.moves import xrange
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from .sparsefuncs_fast import csr_row_norms
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from .validation import check_array
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@deprecated("sklearn.utils.extmath.norm was deprecated in version 0.19 "
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"and will be removed in 0.21. Use scipy.linalg.norm instead.")
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def norm(x):
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"""Compute the Euclidean or Frobenius norm of x.
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Returns the Euclidean norm when x is a vector, the Frobenius norm when x
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is a matrix (2-d array). More precise than sqrt(squared_norm(x)).
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"""
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return linalg.norm(x)
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def squared_norm(x):
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"""Squared Euclidean or Frobenius norm of x.
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Returns the Euclidean norm when x is a vector, the Frobenius norm when x
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is a matrix (2-d array). Faster than norm(x) ** 2.
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"""
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x = np.ravel(x, order='K')
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if np.issubdtype(x.dtype, np.integer):
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warnings.warn('Array type is integer, np.dot may overflow. '
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'Data should be float type to avoid this issue',
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UserWarning)
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return np.dot(x, x)
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def row_norms(X, squared=False):
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"""Row-wise (squared) Euclidean norm of X.
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Equivalent to np.sqrt((X * X).sum(axis=1)), but also supports sparse
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matrices and does not create an X.shape-sized temporary.
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Performs no input validation.
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"""
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if issparse(X):
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if not isinstance(X, csr_matrix):
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X = csr_matrix(X)
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norms = csr_row_norms(X)
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else:
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norms = np.einsum('ij,ij->i', X, X)
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if not squared:
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np.sqrt(norms, norms)
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return norms
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def fast_logdet(A):
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"""Compute log(det(A)) for A symmetric
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Equivalent to : np.log(nl.det(A)) but more robust.
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It returns -Inf if det(A) is non positive or is not defined.
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"""
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sign, ld = np.linalg.slogdet(A)
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if not sign > 0:
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return -np.inf
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return ld
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def _impose_f_order(X):
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"""Helper Function"""
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# important to access flags instead of calling np.isfortran,
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# this catches corner cases.
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if X.flags.c_contiguous:
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return check_array(X.T, copy=False, order='F'), True
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else:
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return check_array(X, copy=False, order='F'), False
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@deprecated("sklearn.utils.extmath.fast_dot was deprecated in version 0.19 "
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"and will be removed in 0.21. Use the equivalent np.dot instead.")
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def fast_dot(a, b, out=None):
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return np.dot(a, b, out)
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def density(w, **kwargs):
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"""Compute density of a sparse vector
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Return a value between 0 and 1
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"""
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if hasattr(w, "toarray"):
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d = float(w.nnz) / (w.shape[0] * w.shape[1])
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else:
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d = 0 if w is None else float((w != 0).sum()) / w.size
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return d
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def safe_sparse_dot(a, b, dense_output=False):
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"""Dot product that handle the sparse matrix case correctly
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Uses BLAS GEMM as replacement for numpy.dot where possible
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to avoid unnecessary copies.
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Parameters
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----------
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a : array or sparse matrix
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b : array or sparse matrix
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dense_output : boolean, default False
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When False, either ``a`` or ``b`` being sparse will yield sparse
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output. When True, output will always be an array.
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Returns
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-------
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dot_product : array or sparse matrix
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sparse if ``a`` or ``b`` is sparse and ``dense_output=False``.
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"""
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if issparse(a) or issparse(b):
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ret = a * b
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if dense_output and hasattr(ret, "toarray"):
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ret = ret.toarray()
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return ret
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else:
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return np.dot(a, b)
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def randomized_range_finder(A, size, n_iter,
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power_iteration_normalizer='auto',
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random_state=None):
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"""Computes an orthonormal matrix whose range approximates the range of A.
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Parameters
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----------
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A : 2D array
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The input data matrix
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size : integer
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Size of the return array
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n_iter : integer
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Number of power iterations used to stabilize the result
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power_iteration_normalizer : 'auto' (default), 'QR', 'LU', 'none'
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Whether the power iterations are normalized with step-by-step
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QR factorization (the slowest but most accurate), 'none'
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(the fastest but numerically unstable when `n_iter` is large, e.g.
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typically 5 or larger), or 'LU' factorization (numerically stable
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but can lose slightly in accuracy). The 'auto' mode applies no
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normalization if `n_iter`<=2 and switches to LU otherwise.
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.. versionadded:: 0.18
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random_state : int, RandomState instance or None, optional (default=None)
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The seed of the pseudo random number generator to use when shuffling
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the data. If int, random_state is the seed used by the random number
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generator; If RandomState instance, random_state is the random number
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generator; If None, the random number generator is the RandomState
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instance used by `np.random`.
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Returns
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-------
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Q : 2D array
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A (size x size) projection matrix, the range of which
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approximates well the range of the input matrix A.
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Notes
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-----
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Follows Algorithm 4.3 of
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Finding structure with randomness: Stochastic algorithms for constructing
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approximate matrix decompositions
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Halko, et al., 2009 (arXiv:909) http://arxiv.org/pdf/0909.4061
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An implementation of a randomized algorithm for principal component
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analysis
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A. Szlam et al. 2014
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"""
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random_state = check_random_state(random_state)
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# Generating normal random vectors with shape: (A.shape[1], size)
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Q = random_state.normal(size=(A.shape[1], size))
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if A.dtype.kind == 'f':
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# Ensure f32 is preserved as f32
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Q = Q.astype(A.dtype, copy=False)
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# Deal with "auto" mode
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if power_iteration_normalizer == 'auto':
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if n_iter <= 2:
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power_iteration_normalizer = 'none'
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else:
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power_iteration_normalizer = 'LU'
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# Perform power iterations with Q to further 'imprint' the top
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# singular vectors of A in Q
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for i in range(n_iter):
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if power_iteration_normalizer == 'none':
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Q = safe_sparse_dot(A, Q)
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Q = safe_sparse_dot(A.T, Q)
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elif power_iteration_normalizer == 'LU':
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Q, _ = linalg.lu(safe_sparse_dot(A, Q), permute_l=True)
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Q, _ = linalg.lu(safe_sparse_dot(A.T, Q), permute_l=True)
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elif power_iteration_normalizer == 'QR':
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Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode='economic')
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Q, _ = linalg.qr(safe_sparse_dot(A.T, Q), mode='economic')
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# Sample the range of A using by linear projection of Q
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# Extract an orthonormal basis
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Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode='economic')
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return Q
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def randomized_svd(M, n_components, n_oversamples=10, n_iter='auto',
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power_iteration_normalizer='auto', transpose='auto',
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flip_sign=True, random_state=0):
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"""Computes a truncated randomized SVD
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Parameters
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----------
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M : ndarray or sparse matrix
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Matrix to decompose
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n_components : int
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Number of singular values and vectors to extract.
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n_oversamples : int (default is 10)
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Additional number of random vectors to sample the range of M so as
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to ensure proper conditioning. The total number of random vectors
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used to find the range of M is n_components + n_oversamples. Smaller
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number can improve speed but can negatively impact the quality of
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approximation of singular vectors and singular values.
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n_iter : int or 'auto' (default is 'auto')
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Number of power iterations. It can be used to deal with very noisy
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problems. When 'auto', it is set to 4, unless `n_components` is small
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(< .1 * min(X.shape)) `n_iter` in which case is set to 7.
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This improves precision with few components.
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.. versionchanged:: 0.18
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power_iteration_normalizer : 'auto' (default), 'QR', 'LU', 'none'
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Whether the power iterations are normalized with step-by-step
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QR factorization (the slowest but most accurate), 'none'
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(the fastest but numerically unstable when `n_iter` is large, e.g.
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typically 5 or larger), or 'LU' factorization (numerically stable
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but can lose slightly in accuracy). The 'auto' mode applies no
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normalization if `n_iter`<=2 and switches to LU otherwise.
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.. versionadded:: 0.18
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transpose : True, False or 'auto' (default)
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Whether the algorithm should be applied to M.T instead of M. The
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result should approximately be the same. The 'auto' mode will
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trigger the transposition if M.shape[1] > M.shape[0] since this
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implementation of randomized SVD tend to be a little faster in that
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case.
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.. versionchanged:: 0.18
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flip_sign : boolean, (True by default)
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The output of a singular value decomposition is only unique up to a
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permutation of the signs of the singular vectors. If `flip_sign` is
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set to `True`, the sign ambiguity is resolved by making the largest
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loadings for each component in the left singular vectors positive.
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random_state : int, RandomState instance or None, optional (default=None)
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The seed of the pseudo random number generator to use when shuffling
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the data. If int, random_state is the seed used by the random number
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generator; If RandomState instance, random_state is the random number
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generator; If None, the random number generator is the RandomState
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instance used by `np.random`.
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Notes
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-----
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This algorithm finds a (usually very good) approximate truncated
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singular value decomposition using randomization to speed up the
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computations. It is particularly fast on large matrices on which
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you wish to extract only a small number of components. In order to
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obtain further speed up, `n_iter` can be set <=2 (at the cost of
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loss of precision).
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References
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----------
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* Finding structure with randomness: Stochastic algorithms for constructing
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approximate matrix decompositions
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Halko, et al., 2009 http://arxiv.org/abs/arXiv:0909.4061
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* A randomized algorithm for the decomposition of matrices
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Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
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* An implementation of a randomized algorithm for principal component
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analysis
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A. Szlam et al. 2014
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"""
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random_state = check_random_state(random_state)
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n_random = n_components + n_oversamples
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n_samples, n_features = M.shape
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if n_iter == 'auto':
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# Checks if the number of iterations is explicitly specified
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# Adjust n_iter. 7 was found a good compromise for PCA. See #5299
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n_iter = 7 if n_components < .1 * min(M.shape) else 4
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if transpose == 'auto':
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transpose = n_samples < n_features
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if transpose:
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# this implementation is a bit faster with smaller shape[1]
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M = M.T
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Q = randomized_range_finder(M, n_random, n_iter,
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power_iteration_normalizer, random_state)
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# project M to the (k + p) dimensional space using the basis vectors
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B = safe_sparse_dot(Q.T, M)
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# compute the SVD on the thin matrix: (k + p) wide
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Uhat, s, V = linalg.svd(B, full_matrices=False)
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del B
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U = np.dot(Q, Uhat)
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if flip_sign:
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if not transpose:
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U, V = svd_flip(U, V)
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else:
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# In case of transpose u_based_decision=false
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# to actually flip based on u and not v.
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U, V = svd_flip(U, V, u_based_decision=False)
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if transpose:
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# transpose back the results according to the input convention
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return V[:n_components, :].T, s[:n_components], U[:, :n_components].T
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else:
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return U[:, :n_components], s[:n_components], V[:n_components, :]
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@deprecated("sklearn.utils.extmath.logsumexp was deprecated in version 0.19 "
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"and will be removed in 0.21. Use scipy.misc.logsumexp instead.")
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def logsumexp(arr, axis=0):
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"""Computes the sum of arr assuming arr is in the log domain.
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Returns log(sum(exp(arr))) while minimizing the possibility of
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over/underflow.
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Examples
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--------
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>>> import numpy as np
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>>> from sklearn.utils.extmath import logsumexp
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>>> a = np.arange(10)
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>>> np.log(np.sum(np.exp(a)))
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9.4586297444267107
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>>> logsumexp(a)
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9.4586297444267107
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"""
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return scipy_logsumexp(arr, axis)
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def weighted_mode(a, w, axis=0):
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"""Returns an array of the weighted modal (most common) value in a
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If there is more than one such value, only the first is returned.
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The bin-count for the modal bins is also returned.
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This is an extension of the algorithm in scipy.stats.mode.
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Parameters
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----------
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a : array_like
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n-dimensional array of which to find mode(s).
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w : array_like
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n-dimensional array of weights for each value
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axis : int, optional
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Axis along which to operate. Default is 0, i.e. the first axis.
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Returns
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-------
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vals : ndarray
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Array of modal values.
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score : ndarray
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Array of weighted counts for each mode.
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Examples
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--------
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>>> from sklearn.utils.extmath import weighted_mode
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>>> x = [4, 1, 4, 2, 4, 2]
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>>> weights = [1, 1, 1, 1, 1, 1]
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>>> weighted_mode(x, weights)
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(array([ 4.]), array([ 3.]))
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The value 4 appears three times: with uniform weights, the result is
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simply the mode of the distribution.
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>>> weights = [1, 3, 0.5, 1.5, 1, 2] # deweight the 4's
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>>> weighted_mode(x, weights)
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(array([ 2.]), array([ 3.5]))
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The value 2 has the highest score: it appears twice with weights of
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1.5 and 2: the sum of these is 3.
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See Also
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--------
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scipy.stats.mode
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"""
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if axis is None:
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a = np.ravel(a)
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w = np.ravel(w)
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axis = 0
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else:
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a = np.asarray(a)
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w = np.asarray(w)
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axis = axis
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if a.shape != w.shape:
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w = np.zeros(a.shape, dtype=w.dtype) + w
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scores = np.unique(np.ravel(a)) # get ALL unique values
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testshape = list(a.shape)
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testshape[axis] = 1
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oldmostfreq = np.zeros(testshape)
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oldcounts = np.zeros(testshape)
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for score in scores:
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template = np.zeros(a.shape)
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ind = (a == score)
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template[ind] = w[ind]
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counts = np.expand_dims(np.sum(template, axis), axis)
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mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
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oldcounts = np.maximum(counts, oldcounts)
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oldmostfreq = mostfrequent
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return mostfrequent, oldcounts
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@deprecated("sklearn.utils.extmath.pinvh was deprecated in version 0.19 "
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"and will be removed in 0.21. Use scipy.linalg.pinvh instead.")
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def pinvh(a, cond=None, rcond=None, lower=True):
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return linalg.pinvh(a, cond, rcond, lower)
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def cartesian(arrays, out=None):
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"""Generate a cartesian product of input arrays.
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Parameters
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----------
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arrays : list of array-like
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1-D arrays to form the cartesian product of.
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out : ndarray
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Array to place the cartesian product in.
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Returns
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-------
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out : ndarray
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2-D array of shape (M, len(arrays)) containing cartesian products
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formed of input arrays.
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Examples
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--------
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>>> cartesian(([1, 2, 3], [4, 5], [6, 7]))
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array([[1, 4, 6],
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[1, 4, 7],
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[1, 5, 6],
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[1, 5, 7],
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[2, 4, 6],
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[2, 4, 7],
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[2, 5, 6],
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[2, 5, 7],
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[3, 4, 6],
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[3, 4, 7],
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[3, 5, 6],
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[3, 5, 7]])
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"""
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arrays = [np.asarray(x) for x in arrays]
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shape = (len(x) for x in arrays)
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dtype = arrays[0].dtype
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ix = np.indices(shape)
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ix = ix.reshape(len(arrays), -1).T
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if out is None:
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out = np.empty_like(ix, dtype=dtype)
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for n, arr in enumerate(arrays):
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out[:, n] = arrays[n][ix[:, n]]
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return out
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def svd_flip(u, v, u_based_decision=True):
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"""Sign correction to ensure deterministic output from SVD.
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Adjusts the columns of u and the rows of v such that the loadings in the
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columns in u that are largest in absolute value are always positive.
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Parameters
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----------
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u, v : ndarray
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u and v are the output of `linalg.svd` or
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`sklearn.utils.extmath.randomized_svd`, with matching inner dimensions
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so one can compute `np.dot(u * s, v)`.
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u_based_decision : boolean, (default=True)
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If True, use the columns of u as the basis for sign flipping.
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Otherwise, use the rows of v. The choice of which variable to base the
|
|
decision on is generally algorithm dependent.
|
|
|
|
|
|
Returns
|
|
-------
|
|
u_adjusted, v_adjusted : arrays with the same dimensions as the input.
|
|
|
|
"""
|
|
if u_based_decision:
|
|
# columns of u, rows of v
|
|
max_abs_cols = np.argmax(np.abs(u), axis=0)
|
|
signs = np.sign(u[max_abs_cols, xrange(u.shape[1])])
|
|
u *= signs
|
|
v *= signs[:, np.newaxis]
|
|
else:
|
|
# rows of v, columns of u
|
|
max_abs_rows = np.argmax(np.abs(v), axis=1)
|
|
signs = np.sign(v[xrange(v.shape[0]), max_abs_rows])
|
|
u *= signs
|
|
v *= signs[:, np.newaxis]
|
|
return u, v
|
|
|
|
|
|
def log_logistic(X, out=None):
|
|
"""Compute the log of the logistic function, ``log(1 / (1 + e ** -x))``.
|
|
|
|
This implementation is numerically stable because it splits positive and
|
|
negative values::
|
|
|
|
-log(1 + exp(-x_i)) if x_i > 0
|
|
x_i - log(1 + exp(x_i)) if x_i <= 0
|
|
|
|
For the ordinary logistic function, use ``scipy.special.expit``.
|
|
|
|
Parameters
|
|
----------
|
|
X : array-like, shape (M, N) or (M, )
|
|
Argument to the logistic function
|
|
|
|
out : array-like, shape: (M, N) or (M, ), optional:
|
|
Preallocated output array.
|
|
|
|
Returns
|
|
-------
|
|
out : array, shape (M, N) or (M, )
|
|
Log of the logistic function evaluated at every point in x
|
|
|
|
Notes
|
|
-----
|
|
See the blog post describing this implementation:
|
|
http://fa.bianp.net/blog/2013/numerical-optimizers-for-logistic-regression/
|
|
"""
|
|
is_1d = X.ndim == 1
|
|
X = np.atleast_2d(X)
|
|
X = check_array(X, dtype=np.float64)
|
|
|
|
n_samples, n_features = X.shape
|
|
|
|
if out is None:
|
|
out = np.empty_like(X)
|
|
|
|
_log_logistic_sigmoid(n_samples, n_features, X, out)
|
|
|
|
if is_1d:
|
|
return np.squeeze(out)
|
|
return out
|
|
|
|
|
|
def softmax(X, copy=True):
|
|
"""
|
|
Calculate the softmax function.
|
|
|
|
The softmax function is calculated by
|
|
np.exp(X) / np.sum(np.exp(X), axis=1)
|
|
|
|
This will cause overflow when large values are exponentiated.
|
|
Hence the largest value in each row is subtracted from each data
|
|
point to prevent this.
|
|
|
|
Parameters
|
|
----------
|
|
X : array-like, shape (M, N)
|
|
Argument to the logistic function
|
|
|
|
copy : bool, optional
|
|
Copy X or not.
|
|
|
|
Returns
|
|
-------
|
|
out : array, shape (M, N)
|
|
Softmax function evaluated at every point in x
|
|
"""
|
|
if copy:
|
|
X = np.copy(X)
|
|
max_prob = np.max(X, axis=1).reshape((-1, 1))
|
|
X -= max_prob
|
|
np.exp(X, X)
|
|
sum_prob = np.sum(X, axis=1).reshape((-1, 1))
|
|
X /= sum_prob
|
|
return X
|
|
|
|
|
|
def safe_min(X):
|
|
"""Returns the minimum value of a dense or a CSR/CSC matrix.
|
|
|
|
Adapated from http://stackoverflow.com/q/13426580
|
|
|
|
"""
|
|
if issparse(X):
|
|
if len(X.data) == 0:
|
|
return 0
|
|
m = X.data.min()
|
|
return m if X.getnnz() == X.size else min(m, 0)
|
|
else:
|
|
return X.min()
|
|
|
|
|
|
def make_nonnegative(X, min_value=0):
|
|
"""Ensure `X.min()` >= `min_value`."""
|
|
min_ = safe_min(X)
|
|
if min_ < min_value:
|
|
if issparse(X):
|
|
raise ValueError("Cannot make the data matrix"
|
|
" nonnegative because it is sparse."
|
|
" Adding a value to every entry would"
|
|
" make it no longer sparse.")
|
|
X = X + (min_value - min_)
|
|
return X
|
|
|
|
|
|
def _incremental_mean_and_var(X, last_mean=.0, last_variance=None,
|
|
last_sample_count=0):
|
|
"""Calculate mean update and a Youngs and Cramer variance update.
|
|
|
|
last_mean and last_variance are statistics computed at the last step by the
|
|
function. Both must be initialized to 0.0. In case no scaling is required
|
|
last_variance can be None. The mean is always required and returned because
|
|
necessary for the calculation of the variance. last_n_samples_seen is the
|
|
number of samples encountered until now.
|
|
|
|
From the paper "Algorithms for computing the sample variance: analysis and
|
|
recommendations", by Chan, Golub, and LeVeque.
|
|
|
|
Parameters
|
|
----------
|
|
X : array-like, shape (n_samples, n_features)
|
|
Data to use for variance update
|
|
|
|
last_mean : array-like, shape: (n_features,)
|
|
|
|
last_variance : array-like, shape: (n_features,)
|
|
|
|
last_sample_count : int
|
|
|
|
Returns
|
|
-------
|
|
updated_mean : array, shape (n_features,)
|
|
|
|
updated_variance : array, shape (n_features,)
|
|
If None, only mean is computed
|
|
|
|
updated_sample_count : int
|
|
|
|
References
|
|
----------
|
|
T. Chan, G. Golub, R. LeVeque. Algorithms for computing the sample
|
|
variance: recommendations, The American Statistician, Vol. 37, No. 3,
|
|
pp. 242-247
|
|
|
|
Also, see the sparse implementation of this in
|
|
`utils.sparsefuncs.incr_mean_variance_axis` and
|
|
`utils.sparsefuncs_fast.incr_mean_variance_axis0`
|
|
"""
|
|
# old = stats until now
|
|
# new = the current increment
|
|
# updated = the aggregated stats
|
|
last_sum = last_mean * last_sample_count
|
|
new_sum = X.sum(axis=0)
|
|
|
|
new_sample_count = X.shape[0]
|
|
updated_sample_count = last_sample_count + new_sample_count
|
|
|
|
updated_mean = (last_sum + new_sum) / updated_sample_count
|
|
|
|
if last_variance is None:
|
|
updated_variance = None
|
|
else:
|
|
new_unnormalized_variance = X.var(axis=0) * new_sample_count
|
|
if last_sample_count == 0: # Avoid division by 0
|
|
updated_unnormalized_variance = new_unnormalized_variance
|
|
else:
|
|
last_over_new_count = last_sample_count / new_sample_count
|
|
last_unnormalized_variance = last_variance * last_sample_count
|
|
updated_unnormalized_variance = (
|
|
last_unnormalized_variance +
|
|
new_unnormalized_variance +
|
|
last_over_new_count / updated_sample_count *
|
|
(last_sum / last_over_new_count - new_sum) ** 2)
|
|
updated_variance = updated_unnormalized_variance / updated_sample_count
|
|
|
|
return updated_mean, updated_variance, updated_sample_count
|
|
|
|
|
|
def _deterministic_vector_sign_flip(u):
|
|
"""Modify the sign of vectors for reproducibility
|
|
|
|
Flips the sign of elements of all the vectors (rows of u) such that
|
|
the absolute maximum element of each vector is positive.
|
|
|
|
Parameters
|
|
----------
|
|
u : ndarray
|
|
Array with vectors as its rows.
|
|
|
|
Returns
|
|
-------
|
|
u_flipped : ndarray with same shape as u
|
|
Array with the sign flipped vectors as its rows.
|
|
"""
|
|
max_abs_rows = np.argmax(np.abs(u), axis=1)
|
|
signs = np.sign(u[range(u.shape[0]), max_abs_rows])
|
|
u *= signs[:, np.newaxis]
|
|
return u
|
|
|
|
|
|
def stable_cumsum(arr, axis=None, rtol=1e-05, atol=1e-08):
|
|
"""Use high precision for cumsum and check that final value matches sum
|
|
|
|
Parameters
|
|
----------
|
|
arr : array-like
|
|
To be cumulatively summed as flat
|
|
axis : int, optional
|
|
Axis along which the cumulative sum is computed.
|
|
The default (None) is to compute the cumsum over the flattened array.
|
|
rtol : float
|
|
Relative tolerance, see ``np.allclose``
|
|
atol : float
|
|
Absolute tolerance, see ``np.allclose``
|
|
"""
|
|
# sum is as unstable as cumsum for numpy < 1.9
|
|
if np_version < (1, 9):
|
|
return np.cumsum(arr, axis=axis, dtype=np.float64)
|
|
|
|
out = np.cumsum(arr, axis=axis, dtype=np.float64)
|
|
expected = np.sum(arr, axis=axis, dtype=np.float64)
|
|
if not np.all(np.isclose(out.take(-1, axis=axis), expected, rtol=rtol,
|
|
atol=atol, equal_nan=True)):
|
|
warnings.warn('cumsum was found to be unstable: '
|
|
'its last element does not correspond to sum',
|
|
RuntimeWarning)
|
|
return out
|