1151 lines
38 KiB
Python
1151 lines
38 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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import warnings
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import numpy as np
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from scipy import linalg, sparse
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from . import check_random_state
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from ._logistic_sigmoid import _log_logistic_sigmoid
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from .sparsefuncs_fast import csr_row_norms
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from .validation import check_array
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from ._array_api import get_namespace
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def squared_norm(x):
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"""Squared Euclidean or Frobenius norm of x.
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Faster than norm(x) ** 2.
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Parameters
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----------
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x : array-like
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The input array which could be either be a vector or a 2 dimensional array.
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Returns
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-------
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float
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The Euclidean norm when x is a vector, the Frobenius norm when x
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is a matrix (2-d array).
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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(
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"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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)
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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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Parameters
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----------
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X : array-like
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The input array.
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squared : bool, default=False
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If True, return squared norms.
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Returns
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-------
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array-like
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The row-wise (squared) Euclidean norm of X.
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"""
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if sparse.issparse(X):
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if not isinstance(X, sparse.csr_matrix):
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X = sparse.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 logarithm of determinant of a square matrix.
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The (natural) logarithm of the determinant of a square matrix
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is returned if det(A) is non-negative and well defined.
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If the determinant is zero or negative returns -Inf.
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Equivalent to : np.log(np.det(A)) but more robust.
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Parameters
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----------
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A : array_like of shape (n, n)
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The square matrix.
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Returns
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-------
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logdet : float
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When det(A) is strictly positive, log(det(A)) is returned.
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When det(A) is non-positive or not defined, then -inf is returned.
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See Also
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--------
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numpy.linalg.slogdet : Compute the sign and (natural) logarithm of the determinant
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of an array.
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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 fast_logdet
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>>> a = np.array([[5, 1], [2, 8]])
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>>> fast_logdet(a)
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3.6375861597263857
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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 density(w, **kwargs):
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"""Compute density of a sparse vector.
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Parameters
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----------
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w : array-like
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The sparse vector.
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**kwargs : keyword arguments
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Ignored.
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.. deprecated:: 1.2
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``**kwargs`` were deprecated in version 1.2 and will be removed in
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1.4.
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Returns
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-------
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float
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The density of w, between 0 and 1.
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"""
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if kwargs:
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warnings.warn(
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"Additional keyword arguments are deprecated in version 1.2 and will be"
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" removed in version 1.4.",
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FutureWarning,
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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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Parameters
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----------
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a : {ndarray, sparse matrix}
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b : {ndarray, sparse matrix}
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dense_output : bool, default=False
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When False, ``a`` and ``b`` both being sparse will yield sparse output.
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When True, output will always be a dense array.
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Returns
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-------
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dot_product : {ndarray, sparse matrix}
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Sparse if ``a`` and ``b`` are sparse and ``dense_output=False``.
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"""
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if a.ndim > 2 or b.ndim > 2:
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if sparse.issparse(a):
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# sparse is always 2D. Implies b is 3D+
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# [i, j] @ [k, ..., l, m, n] -> [i, k, ..., l, n]
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b_ = np.rollaxis(b, -2)
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b_2d = b_.reshape((b.shape[-2], -1))
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ret = a @ b_2d
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ret = ret.reshape(a.shape[0], *b_.shape[1:])
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elif sparse.issparse(b):
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# sparse is always 2D. Implies a is 3D+
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# [k, ..., l, m] @ [i, j] -> [k, ..., l, j]
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a_2d = a.reshape(-1, a.shape[-1])
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ret = a_2d @ b
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ret = ret.reshape(*a.shape[:-1], b.shape[1])
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else:
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ret = np.dot(a, b)
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else:
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ret = a @ b
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if (
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sparse.issparse(a)
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and sparse.issparse(b)
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and dense_output
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and hasattr(ret, "toarray")
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):
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return ret.toarray()
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return ret
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def randomized_range_finder(
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A, *, size, n_iter, power_iteration_normalizer="auto", random_state=None
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):
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"""Compute 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 : int
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Size of the return array.
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n_iter : int
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Number of power iterations used to stabilize the result.
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power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'
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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, default=None
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The seed of the pseudo random number generator to use when shuffling
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the data, i.e. getting the random vectors to initialize the algorithm.
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Pass an int for reproducible results across multiple function calls.
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See :term:`Glossary <random_state>`.
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Returns
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-------
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Q : ndarray
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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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:arxiv:`"Finding structure with randomness:
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Stochastic algorithms for constructing approximate matrix decompositions"
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<0909.4061>`
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Halko, et al. (2009)
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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 hasattr(A, "dtype") and 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(
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M,
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n_components,
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*,
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n_oversamples=10,
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n_iter="auto",
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power_iteration_normalizer="auto",
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transpose="auto",
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flip_sign=True,
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random_state=None,
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svd_lapack_driver="gesdd",
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):
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"""Compute a truncated randomized SVD.
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This method solves the fixed-rank approximation problem described in [1]_
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(problem (1.5), p5).
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Parameters
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----------
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M : {ndarray, 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=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. Users might wish
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to increase this parameter up to `2*k - n_components` where k is the
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effective rank, for large matrices, noisy problems, matrices with
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slowly decaying spectrums, or to increase precision accuracy. See [1]_
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(pages 5, 23 and 26).
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n_iter : int or 'auto', default='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)) in which case `n_iter` is set to 7.
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This improves precision with few components. Note that in general
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users should rather increase `n_oversamples` before increasing `n_iter`
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as the principle of the randomized method is to avoid usage of these
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more costly power iterations steps. When `n_components` is equal
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or greater to the effective matrix rank and the spectrum does not
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present a slow decay, `n_iter=0` or `1` should even work fine in theory
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(see [1]_ page 9).
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.. versionchanged:: 0.18
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power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'
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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'
|
|
(the fastest but numerically unstable when `n_iter` is large, e.g.
|
|
typically 5 or larger), or 'LU' factorization (numerically stable
|
|
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 : bool or 'auto', default='auto'
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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 : bool, default=True
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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, default='warn'
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The seed of the pseudo random number generator to use when
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shuffling the data, i.e. getting the random vectors to initialize
|
|
the algorithm. Pass an int for reproducible results across multiple
|
|
function calls. See :term:`Glossary <random_state>`.
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|
|
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.. versionchanged:: 1.2
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The default value changed from 0 to None.
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svd_lapack_driver : {"gesdd", "gesvd"}, default="gesdd"
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Whether to use the more efficient divide-and-conquer approach
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(`"gesdd"`) or more general rectangular approach (`"gesvd"`) to compute
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the SVD of the matrix B, which is the projection of M into a low
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dimensional subspace, as described in [1]_.
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.. versionadded:: 1.2
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Returns
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-------
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u : ndarray of shape (n_samples, n_components)
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Unitary matrix having left singular vectors with signs flipped as columns.
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s : ndarray of shape (n_components,)
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The singular values, sorted in non-increasing order.
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vh : ndarray of shape (n_components, n_features)
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Unitary matrix having right singular vectors with signs flipped as rows.
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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). To increase the precision it is recommended to
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increase `n_oversamples`, up to `2*k-n_components` where k is the
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effective rank. Usually, `n_components` is chosen to be greater than k
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so increasing `n_oversamples` up to `n_components` should be enough.
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References
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----------
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.. [1] :arxiv:`"Finding structure with randomness:
|
|
Stochastic algorithms for constructing approximate matrix decompositions"
|
|
<0909.4061>`
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|
Halko, et al. (2009)
|
|
|
|
.. [2] A randomized algorithm for the decomposition of matrices
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Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
|
|
|
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.. [3] An implementation of a randomized algorithm for principal component
|
|
analysis A. Szlam et al. 2014
|
|
|
|
Examples
|
|
--------
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>>> import numpy as np
|
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>>> from sklearn.utils.extmath import randomized_svd
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>>> a = np.array([[1, 2, 3, 5],
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... [3, 4, 5, 6],
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... [7, 8, 9, 10]])
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>>> U, s, Vh = randomized_svd(a, n_components=2, random_state=0)
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>>> U.shape, s.shape, Vh.shape
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((3, 2), (2,), (2, 4))
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"""
|
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if isinstance(M, (sparse.lil_matrix, sparse.dok_matrix)):
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warnings.warn(
|
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"Calculating SVD of a {} is expensive. "
|
|
"csr_matrix is more efficient.".format(type(M).__name__),
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sparse.SparseEfficiencyWarning,
|
|
)
|
|
|
|
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":
|
|
# 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
|
|
n_iter = 7 if n_components < 0.1 * min(M.shape) else 4
|
|
|
|
if transpose == "auto":
|
|
transpose = n_samples < n_features
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|
if transpose:
|
|
# this implementation is a bit faster with smaller shape[1]
|
|
M = M.T
|
|
|
|
Q = randomized_range_finder(
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M,
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size=n_random,
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n_iter=n_iter,
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power_iteration_normalizer=power_iteration_normalizer,
|
|
random_state=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)
|
|
|
|
# compute the SVD on the thin matrix: (k + p) wide
|
|
Uhat, s, Vt = linalg.svd(B, full_matrices=False, lapack_driver=svd_lapack_driver)
|
|
|
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del B
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U = np.dot(Q, Uhat)
|
|
|
|
if flip_sign:
|
|
if not transpose:
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|
U, Vt = svd_flip(U, Vt)
|
|
else:
|
|
# In case of transpose u_based_decision=false
|
|
# to actually flip based on u and not v.
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|
U, Vt = svd_flip(U, Vt, u_based_decision=False)
|
|
|
|
if transpose:
|
|
# transpose back the results according to the input convention
|
|
return Vt[:n_components, :].T, s[:n_components], U[:, :n_components].T
|
|
else:
|
|
return U[:, :n_components], s[:n_components], Vt[:n_components, :]
|
|
|
|
|
|
def _randomized_eigsh(
|
|
M,
|
|
n_components,
|
|
*,
|
|
n_oversamples=10,
|
|
n_iter="auto",
|
|
power_iteration_normalizer="auto",
|
|
selection="module",
|
|
random_state=None,
|
|
):
|
|
"""Computes a truncated eigendecomposition using randomized methods
|
|
|
|
This method solves the fixed-rank approximation problem described in the
|
|
Halko et al paper.
|
|
|
|
The choice of which components to select can be tuned with the `selection`
|
|
parameter.
|
|
|
|
.. versionadded:: 0.24
|
|
|
|
Parameters
|
|
----------
|
|
M : ndarray or sparse matrix
|
|
Matrix to decompose, it should be real symmetric square or complex
|
|
hermitian
|
|
|
|
n_components : int
|
|
Number of eigenvalues and vectors to extract.
|
|
|
|
n_oversamples : int, default=10
|
|
Additional number of random vectors to sample the range of M so as
|
|
to ensure proper conditioning. The total number of random vectors
|
|
used to find the range of M is n_components + n_oversamples. Smaller
|
|
number can improve speed but can negatively impact the quality of
|
|
approximation of eigenvectors and eigenvalues. Users might wish
|
|
to increase this parameter up to `2*k - n_components` where k is the
|
|
effective rank, for large matrices, noisy problems, matrices with
|
|
slowly decaying spectrums, or to increase precision accuracy. See Halko
|
|
et al (pages 5, 23 and 26).
|
|
|
|
n_iter : int or 'auto', default='auto'
|
|
Number of power iterations. It can be used to deal with very noisy
|
|
problems. When 'auto', it is set to 4, unless `n_components` is small
|
|
(< .1 * min(X.shape)) in which case `n_iter` is set to 7.
|
|
This improves precision with few components. Note that in general
|
|
users should rather increase `n_oversamples` before increasing `n_iter`
|
|
as the principle of the randomized method is to avoid usage of these
|
|
more costly power iterations steps. When `n_components` is equal
|
|
or greater to the effective matrix rank and the spectrum does not
|
|
present a slow decay, `n_iter=0` or `1` should even work fine in theory
|
|
(see Halko et al paper, page 9).
|
|
|
|
power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'
|
|
Whether the power iterations are normalized with step-by-step
|
|
QR factorization (the slowest but most accurate), 'none'
|
|
(the fastest but numerically unstable when `n_iter` is large, e.g.
|
|
typically 5 or larger), or 'LU' factorization (numerically stable
|
|
but can lose slightly in accuracy). The 'auto' mode applies no
|
|
normalization if `n_iter` <= 2 and switches to LU otherwise.
|
|
|
|
selection : {'value', 'module'}, default='module'
|
|
Strategy used to select the n components. When `selection` is `'value'`
|
|
(not yet implemented, will become the default when implemented), the
|
|
components corresponding to the n largest eigenvalues are returned.
|
|
When `selection` is `'module'`, the components corresponding to the n
|
|
eigenvalues with largest modules are returned.
|
|
|
|
random_state : int, RandomState instance, default=None
|
|
The seed of the pseudo random number generator to use when shuffling
|
|
the data, i.e. getting the random vectors to initialize the algorithm.
|
|
Pass an int for reproducible results across multiple function calls.
|
|
See :term:`Glossary <random_state>`.
|
|
|
|
Notes
|
|
-----
|
|
This algorithm finds a (usually very good) approximate truncated
|
|
eigendecomposition using randomized methods to speed up the computations.
|
|
|
|
This method is particularly fast on large matrices on which
|
|
you wish to extract only a small number of components. In order to
|
|
obtain further speed up, `n_iter` can be set <=2 (at the cost of
|
|
loss of precision). To increase the precision it is recommended to
|
|
increase `n_oversamples`, up to `2*k-n_components` where k is the
|
|
effective rank. Usually, `n_components` is chosen to be greater than k
|
|
so increasing `n_oversamples` up to `n_components` should be enough.
|
|
|
|
Strategy 'value': not implemented yet.
|
|
Algorithms 5.3, 5.4 and 5.5 in the Halko et al paper should provide good
|
|
condidates for a future implementation.
|
|
|
|
Strategy 'module':
|
|
The principle is that for diagonalizable matrices, the singular values and
|
|
eigenvalues are related: if t is an eigenvalue of A, then :math:`|t|` is a
|
|
singular value of A. This method relies on a randomized SVD to find the n
|
|
singular components corresponding to the n singular values with largest
|
|
modules, and then uses the signs of the singular vectors to find the true
|
|
sign of t: if the sign of left and right singular vectors are different
|
|
then the corresponding eigenvalue is negative.
|
|
|
|
Returns
|
|
-------
|
|
eigvals : 1D array of shape (n_components,) containing the `n_components`
|
|
eigenvalues selected (see ``selection`` parameter).
|
|
eigvecs : 2D array of shape (M.shape[0], n_components) containing the
|
|
`n_components` eigenvectors corresponding to the `eigvals`, in the
|
|
corresponding order. Note that this follows the `scipy.linalg.eigh`
|
|
convention.
|
|
|
|
See Also
|
|
--------
|
|
:func:`randomized_svd`
|
|
|
|
References
|
|
----------
|
|
* :arxiv:`"Finding structure with randomness:
|
|
Stochastic algorithms for constructing approximate matrix decompositions"
|
|
(Algorithm 4.3 for strategy 'module') <0909.4061>`
|
|
Halko, et al. (2009)
|
|
"""
|
|
if selection == "value": # pragma: no cover
|
|
# to do : an algorithm can be found in the Halko et al reference
|
|
raise NotImplementedError()
|
|
|
|
elif selection == "module":
|
|
# Note: no need for deterministic U and Vt (flip_sign=True),
|
|
# as we only use the dot product UVt afterwards
|
|
U, S, Vt = randomized_svd(
|
|
M,
|
|
n_components=n_components,
|
|
n_oversamples=n_oversamples,
|
|
n_iter=n_iter,
|
|
power_iteration_normalizer=power_iteration_normalizer,
|
|
flip_sign=False,
|
|
random_state=random_state,
|
|
)
|
|
|
|
eigvecs = U[:, :n_components]
|
|
eigvals = S[:n_components]
|
|
|
|
# Conversion of Singular values into Eigenvalues:
|
|
# For any eigenvalue t, the corresponding singular value is |t|.
|
|
# So if there is a negative eigenvalue t, the corresponding singular
|
|
# value will be -t, and the left (U) and right (V) singular vectors
|
|
# will have opposite signs.
|
|
# Fastest way: see <https://stackoverflow.com/a/61974002/7262247>
|
|
diag_VtU = np.einsum("ji,ij->j", Vt[:n_components, :], U[:, :n_components])
|
|
signs = np.sign(diag_VtU)
|
|
eigvals = eigvals * signs
|
|
|
|
else: # pragma: no cover
|
|
raise ValueError("Invalid `selection`: %r" % selection)
|
|
|
|
return eigvals, eigvecs
|
|
|
|
|
|
def weighted_mode(a, w, *, axis=0):
|
|
"""Return an array of the weighted modal (most common) value in the passed array.
|
|
|
|
If there is more than one such value, only the first is returned.
|
|
The bin-count for the modal bins is also returned.
|
|
|
|
This is an extension of the algorithm in scipy.stats.mode.
|
|
|
|
Parameters
|
|
----------
|
|
a : array-like of shape (n_samples,)
|
|
Array of which values to find mode(s).
|
|
w : array-like of shape (n_samples,)
|
|
Array of weights for each value.
|
|
axis : int, default=0
|
|
Axis along which to operate. Default is 0, i.e. the first axis.
|
|
|
|
Returns
|
|
-------
|
|
vals : ndarray
|
|
Array of modal values.
|
|
score : ndarray
|
|
Array of weighted counts for each mode.
|
|
|
|
See Also
|
|
--------
|
|
scipy.stats.mode: Calculates the Modal (most common) value of array elements
|
|
along specified axis.
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.utils.extmath import weighted_mode
|
|
>>> x = [4, 1, 4, 2, 4, 2]
|
|
>>> weights = [1, 1, 1, 1, 1, 1]
|
|
>>> weighted_mode(x, weights)
|
|
(array([4.]), array([3.]))
|
|
|
|
The value 4 appears three times: with uniform weights, the result is
|
|
simply the mode of the distribution.
|
|
|
|
>>> weights = [1, 3, 0.5, 1.5, 1, 2] # deweight the 4's
|
|
>>> weighted_mode(x, weights)
|
|
(array([2.]), array([3.5]))
|
|
|
|
The value 2 has the highest score: it appears twice with weights of
|
|
1.5 and 2: the sum of these is 3.5.
|
|
"""
|
|
if axis is None:
|
|
a = np.ravel(a)
|
|
w = np.ravel(w)
|
|
axis = 0
|
|
else:
|
|
a = np.asarray(a)
|
|
w = np.asarray(w)
|
|
|
|
if a.shape != w.shape:
|
|
w = np.full(a.shape, w, dtype=w.dtype)
|
|
|
|
scores = np.unique(np.ravel(a)) # get ALL unique values
|
|
testshape = list(a.shape)
|
|
testshape[axis] = 1
|
|
oldmostfreq = np.zeros(testshape)
|
|
oldcounts = np.zeros(testshape)
|
|
for score in scores:
|
|
template = np.zeros(a.shape)
|
|
ind = a == score
|
|
template[ind] = w[ind]
|
|
counts = np.expand_dims(np.sum(template, axis), axis)
|
|
mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
|
|
oldcounts = np.maximum(counts, oldcounts)
|
|
oldmostfreq = mostfrequent
|
|
return mostfrequent, oldcounts
|
|
|
|
|
|
def cartesian(arrays, out=None):
|
|
"""Generate a cartesian product of input arrays.
|
|
|
|
Parameters
|
|
----------
|
|
arrays : list of array-like
|
|
1-D arrays to form the cartesian product of.
|
|
out : ndarray of shape (M, len(arrays)), default=None
|
|
Array to place the cartesian product in.
|
|
|
|
Returns
|
|
-------
|
|
out : ndarray of shape (M, len(arrays))
|
|
Array containing the cartesian products formed of input arrays.
|
|
If not provided, the `dtype` of the output array is set to the most
|
|
permissive `dtype` of the input arrays, according to NumPy type
|
|
promotion.
|
|
|
|
.. versionadded:: 1.2
|
|
Add support for arrays of different types.
|
|
|
|
Notes
|
|
-----
|
|
This function may not be used on more than 32 arrays
|
|
because the underlying numpy functions do not support it.
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.utils.extmath import cartesian
|
|
>>> cartesian(([1, 2, 3], [4, 5], [6, 7]))
|
|
array([[1, 4, 6],
|
|
[1, 4, 7],
|
|
[1, 5, 6],
|
|
[1, 5, 7],
|
|
[2, 4, 6],
|
|
[2, 4, 7],
|
|
[2, 5, 6],
|
|
[2, 5, 7],
|
|
[3, 4, 6],
|
|
[3, 4, 7],
|
|
[3, 5, 6],
|
|
[3, 5, 7]])
|
|
"""
|
|
arrays = [np.asarray(x) for x in arrays]
|
|
shape = (len(x) for x in arrays)
|
|
|
|
ix = np.indices(shape)
|
|
ix = ix.reshape(len(arrays), -1).T
|
|
|
|
if out is None:
|
|
dtype = np.result_type(*arrays) # find the most permissive dtype
|
|
out = np.empty_like(ix, dtype=dtype)
|
|
|
|
for n, arr in enumerate(arrays):
|
|
out[:, n] = arrays[n][ix[:, n]]
|
|
|
|
return out
|
|
|
|
|
|
def svd_flip(u, v, u_based_decision=True):
|
|
"""Sign correction to ensure deterministic output from SVD.
|
|
|
|
Adjusts the columns of u and the rows of v such that the loadings in the
|
|
columns in u that are largest in absolute value are always positive.
|
|
|
|
Parameters
|
|
----------
|
|
u : ndarray
|
|
Parameters u and v are the output of `linalg.svd` or
|
|
:func:`~sklearn.utils.extmath.randomized_svd`, with matching inner
|
|
dimensions so one can compute `np.dot(u * s, v)`.
|
|
|
|
v : ndarray
|
|
Parameters u and v are the output of `linalg.svd` or
|
|
:func:`~sklearn.utils.extmath.randomized_svd`, with matching inner
|
|
dimensions so one can compute `np.dot(u * s, v)`.
|
|
The input v should really be called vt to be consistent with scipy's
|
|
output.
|
|
|
|
u_based_decision : bool, default=True
|
|
If True, use the columns of u as the basis for sign flipping.
|
|
Otherwise, use the rows of v. The choice of which variable to base the
|
|
decision on is generally algorithm dependent.
|
|
|
|
Returns
|
|
-------
|
|
u_adjusted : ndarray
|
|
Array u with adjusted columns and the same dimensions as u.
|
|
|
|
v_adjusted : ndarray
|
|
Array v with adjusted rows and the same dimensions as v.
|
|
"""
|
|
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, range(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[range(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 of shape (M, N) or (M,)
|
|
Argument to the logistic function.
|
|
|
|
out : array-like of shape (M, N) or (M,), default=None
|
|
Preallocated output array.
|
|
|
|
Returns
|
|
-------
|
|
out : ndarray of 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 of float of shape (M, N)
|
|
Argument to the logistic function.
|
|
|
|
copy : bool, default=True
|
|
Copy X or not.
|
|
|
|
Returns
|
|
-------
|
|
out : ndarray of shape (M, N)
|
|
Softmax function evaluated at every point in x.
|
|
"""
|
|
xp, is_array_api = get_namespace(X)
|
|
if copy:
|
|
X = xp.asarray(X, copy=True)
|
|
max_prob = xp.reshape(xp.max(X, axis=1), (-1, 1))
|
|
X -= max_prob
|
|
|
|
if xp.__name__ in {"numpy", "numpy.array_api"}:
|
|
# optimization for NumPy arrays
|
|
np.exp(X, out=np.asarray(X))
|
|
else:
|
|
# array_api does not have `out=`
|
|
X = xp.exp(X)
|
|
|
|
sum_prob = xp.reshape(xp.sum(X, axis=1), (-1, 1))
|
|
X /= sum_prob
|
|
return X
|
|
|
|
|
|
def make_nonnegative(X, min_value=0):
|
|
"""Ensure `X.min()` >= `min_value`.
|
|
|
|
Parameters
|
|
----------
|
|
X : array-like
|
|
The matrix to make non-negative.
|
|
min_value : float, default=0
|
|
The threshold value.
|
|
|
|
Returns
|
|
-------
|
|
array-like
|
|
The thresholded array.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
When X is sparse.
|
|
"""
|
|
min_ = X.min()
|
|
if min_ < min_value:
|
|
if sparse.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
|
|
|
|
|
|
# Use at least float64 for the accumulating functions to avoid precision issue
|
|
# see https://github.com/numpy/numpy/issues/9393. The float64 is also retained
|
|
# as it is in case the float overflows
|
|
def _safe_accumulator_op(op, x, *args, **kwargs):
|
|
"""
|
|
This function provides numpy accumulator functions with a float64 dtype
|
|
when used on a floating point input. This prevents accumulator overflow on
|
|
smaller floating point dtypes.
|
|
|
|
Parameters
|
|
----------
|
|
op : function
|
|
A numpy accumulator function such as np.mean or np.sum.
|
|
x : ndarray
|
|
A numpy array to apply the accumulator function.
|
|
*args : positional arguments
|
|
Positional arguments passed to the accumulator function after the
|
|
input x.
|
|
**kwargs : keyword arguments
|
|
Keyword arguments passed to the accumulator function.
|
|
|
|
Returns
|
|
-------
|
|
result
|
|
The output of the accumulator function passed to this function.
|
|
"""
|
|
if np.issubdtype(x.dtype, np.floating) and x.dtype.itemsize < 8:
|
|
result = op(x, *args, **kwargs, dtype=np.float64)
|
|
else:
|
|
result = op(x, *args, **kwargs)
|
|
return result
|
|
|
|
|
|
def _incremental_mean_and_var(
|
|
X, last_mean, last_variance, last_sample_count, sample_weight=None
|
|
):
|
|
"""Calculate mean update and a Youngs and Cramer variance update.
|
|
|
|
If sample_weight is given, the weighted mean and variance is computed.
|
|
|
|
Update a given mean and (possibly) variance according to new data given
|
|
in X. last_mean is always required to compute the new mean.
|
|
If last_variance is None, no variance is computed and None return for
|
|
updated_variance.
|
|
|
|
From the paper "Algorithms for computing the sample variance: analysis and
|
|
recommendations", by Chan, Golub, and LeVeque.
|
|
|
|
Parameters
|
|
----------
|
|
X : array-like of shape (n_samples, n_features)
|
|
Data to use for variance update.
|
|
|
|
last_mean : array-like of shape (n_features,)
|
|
|
|
last_variance : array-like of shape (n_features,)
|
|
|
|
last_sample_count : array-like of shape (n_features,)
|
|
The number of samples encountered until now if sample_weight is None.
|
|
If sample_weight is not None, this is the sum of sample_weight
|
|
encountered.
|
|
|
|
sample_weight : array-like of shape (n_samples,) or None
|
|
Sample weights. If None, compute the unweighted mean/variance.
|
|
|
|
Returns
|
|
-------
|
|
updated_mean : ndarray of shape (n_features,)
|
|
|
|
updated_variance : ndarray of shape (n_features,)
|
|
None if last_variance was None.
|
|
|
|
updated_sample_count : ndarray of shape (n_features,)
|
|
|
|
Notes
|
|
-----
|
|
NaNs are ignored during the algorithm.
|
|
|
|
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
|
|
X_nan_mask = np.isnan(X)
|
|
if np.any(X_nan_mask):
|
|
sum_op = np.nansum
|
|
else:
|
|
sum_op = np.sum
|
|
if sample_weight is not None:
|
|
# equivalent to np.nansum(X * sample_weight, axis=0)
|
|
# safer because np.float64(X*W) != np.float64(X)*np.float64(W)
|
|
new_sum = _safe_accumulator_op(
|
|
np.matmul, sample_weight, np.where(X_nan_mask, 0, X)
|
|
)
|
|
new_sample_count = _safe_accumulator_op(
|
|
np.sum, sample_weight[:, None] * (~X_nan_mask), axis=0
|
|
)
|
|
else:
|
|
new_sum = _safe_accumulator_op(sum_op, X, axis=0)
|
|
n_samples = X.shape[0]
|
|
new_sample_count = n_samples - np.sum(X_nan_mask, axis=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:
|
|
T = new_sum / new_sample_count
|
|
temp = X - T
|
|
if sample_weight is not None:
|
|
# equivalent to np.nansum((X-T)**2 * sample_weight, axis=0)
|
|
# safer because np.float64(X*W) != np.float64(X)*np.float64(W)
|
|
correction = _safe_accumulator_op(
|
|
np.matmul, sample_weight, np.where(X_nan_mask, 0, temp)
|
|
)
|
|
temp **= 2
|
|
new_unnormalized_variance = _safe_accumulator_op(
|
|
np.matmul, sample_weight, np.where(X_nan_mask, 0, temp)
|
|
)
|
|
else:
|
|
correction = _safe_accumulator_op(sum_op, temp, axis=0)
|
|
temp **= 2
|
|
new_unnormalized_variance = _safe_accumulator_op(sum_op, temp, axis=0)
|
|
|
|
# correction term of the corrected 2 pass algorithm.
|
|
# See "Algorithms for computing the sample variance: analysis
|
|
# and recommendations", by Chan, Golub, and LeVeque.
|
|
new_unnormalized_variance -= correction**2 / new_sample_count
|
|
|
|
last_unnormalized_variance = last_variance * last_sample_count
|
|
|
|
with np.errstate(divide="ignore", invalid="ignore"):
|
|
last_over_new_count = last_sample_count / new_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
|
|
)
|
|
|
|
zeros = last_sample_count == 0
|
|
updated_unnormalized_variance[zeros] = new_unnormalized_variance[zeros]
|
|
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.
|
|
|
|
Warns if the final cumulative sum does not match the sum (up to the chosen
|
|
tolerance).
|
|
|
|
Parameters
|
|
----------
|
|
arr : array-like
|
|
To be cumulatively summed as flat.
|
|
axis : int, default=None
|
|
Axis along which the cumulative sum is computed.
|
|
The default (None) is to compute the cumsum over the flattened array.
|
|
rtol : float, default=1e-05
|
|
Relative tolerance, see ``np.allclose``.
|
|
atol : float, default=1e-08
|
|
Absolute tolerance, see ``np.allclose``.
|
|
|
|
Returns
|
|
-------
|
|
out : ndarray
|
|
Array with the cumulative sums along the chosen axis.
|
|
"""
|
|
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
|