"""
Extended math utilities.
"""
# Authors: Gael Varoquaux
# Alexandre Gramfort
# Alexandre T. Passos
# Olivier Grisel
# Lars Buitinck
# Stefan van der Walt
# Kyle Kastner
# Giorgio Patrini
# License: BSD 3 clause
import warnings
import numpy as np
from scipy import linalg, sparse
from . import check_random_state
from ._logistic_sigmoid import _log_logistic_sigmoid
from .sparsefuncs_fast import csr_row_norms
from .validation import check_array
from ._array_api import get_namespace
def squared_norm(x):
"""Squared Euclidean or Frobenius norm of x.
Faster than norm(x) ** 2.
Parameters
----------
x : array-like
The input array which could be either be a vector or a 2 dimensional array.
Returns
-------
float
The Euclidean norm when x is a vector, the Frobenius norm when x
is a matrix (2-d array).
"""
x = np.ravel(x, order="K")
if np.issubdtype(x.dtype, np.integer):
warnings.warn(
"Array type is integer, np.dot may overflow. "
"Data should be float type to avoid this issue",
UserWarning,
)
return np.dot(x, x)
def row_norms(X, squared=False):
"""Row-wise (squared) Euclidean norm of X.
Equivalent to np.sqrt((X * X).sum(axis=1)), but also supports sparse
matrices and does not create an X.shape-sized temporary.
Performs no input validation.
Parameters
----------
X : array-like
The input array.
squared : bool, default=False
If True, return squared norms.
Returns
-------
array-like
The row-wise (squared) Euclidean norm of X.
"""
if sparse.issparse(X):
if not isinstance(X, sparse.csr_matrix):
X = sparse.csr_matrix(X)
norms = csr_row_norms(X)
else:
norms = np.einsum("ij,ij->i", X, X)
if not squared:
np.sqrt(norms, norms)
return norms
def fast_logdet(A):
"""Compute logarithm of determinant of a square matrix.
The (natural) logarithm of the determinant of a square matrix
is returned if det(A) is non-negative and well defined.
If the determinant is zero or negative returns -Inf.
Equivalent to : np.log(np.det(A)) but more robust.
Parameters
----------
A : array_like of shape (n, n)
The square matrix.
Returns
-------
logdet : float
When det(A) is strictly positive, log(det(A)) is returned.
When det(A) is non-positive or not defined, then -inf is returned.
See Also
--------
numpy.linalg.slogdet : Compute the sign and (natural) logarithm of the determinant
of an array.
Examples
--------
>>> import numpy as np
>>> from sklearn.utils.extmath import fast_logdet
>>> a = np.array([[5, 1], [2, 8]])
>>> fast_logdet(a)
3.6375861597263857
"""
sign, ld = np.linalg.slogdet(A)
if not sign > 0:
return -np.inf
return ld
def density(w, **kwargs):
"""Compute density of a sparse vector.
Parameters
----------
w : array-like
The sparse vector.
**kwargs : keyword arguments
Ignored.
.. deprecated:: 1.2
``**kwargs`` were deprecated in version 1.2 and will be removed in
1.4.
Returns
-------
float
The density of w, between 0 and 1.
"""
if kwargs:
warnings.warn(
"Additional keyword arguments are deprecated in version 1.2 and will be"
" removed in version 1.4.",
FutureWarning,
)
if hasattr(w, "toarray"):
d = float(w.nnz) / (w.shape[0] * w.shape[1])
else:
d = 0 if w is None else float((w != 0).sum()) / w.size
return d
def safe_sparse_dot(a, b, *, dense_output=False):
"""Dot product that handle the sparse matrix case correctly.
Parameters
----------
a : {ndarray, sparse matrix}
b : {ndarray, sparse matrix}
dense_output : bool, default=False
When False, ``a`` and ``b`` both being sparse will yield sparse output.
When True, output will always be a dense array.
Returns
-------
dot_product : {ndarray, sparse matrix}
Sparse if ``a`` and ``b`` are sparse and ``dense_output=False``.
"""
if a.ndim > 2 or b.ndim > 2:
if sparse.issparse(a):
# sparse is always 2D. Implies b is 3D+
# [i, j] @ [k, ..., l, m, n] -> [i, k, ..., l, n]
b_ = np.rollaxis(b, -2)
b_2d = b_.reshape((b.shape[-2], -1))
ret = a @ b_2d
ret = ret.reshape(a.shape[0], *b_.shape[1:])
elif sparse.issparse(b):
# sparse is always 2D. Implies a is 3D+
# [k, ..., l, m] @ [i, j] -> [k, ..., l, j]
a_2d = a.reshape(-1, a.shape[-1])
ret = a_2d @ b
ret = ret.reshape(*a.shape[:-1], b.shape[1])
else:
ret = np.dot(a, b)
else:
ret = a @ b
if (
sparse.issparse(a)
and sparse.issparse(b)
and dense_output
and hasattr(ret, "toarray")
):
return ret.toarray()
return ret
def randomized_range_finder(
A, *, size, n_iter, power_iteration_normalizer="auto", random_state=None
):
"""Compute an orthonormal matrix whose range approximates the range of A.
Parameters
----------
A : 2D array
The input data matrix.
size : int
Size of the return array.
n_iter : int
Number of power iterations used to stabilize the result.
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.
.. versionadded:: 0.18
random_state : int, RandomState instance or None, 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>`.
Returns
-------
Q : ndarray
A (size x size) projection matrix, the range of which
approximates well the range of the input matrix A.
Notes
-----
Follows Algorithm 4.3 of
:arxiv:`"Finding structure with randomness:
Stochastic algorithms for constructing approximate matrix decompositions"
<0909.4061>`
Halko, et al. (2009)
An implementation of a randomized algorithm for principal component
analysis
A. Szlam et al. 2014
"""
random_state = check_random_state(random_state)
# Generating normal random vectors with shape: (A.shape[1], size)
Q = random_state.normal(size=(A.shape[1], size))
if hasattr(A, "dtype") and A.dtype.kind == "f":
# Ensure f32 is preserved as f32
Q = Q.astype(A.dtype, copy=False)
# Deal with "auto" mode
if power_iteration_normalizer == "auto":
if n_iter <= 2:
power_iteration_normalizer = "none"
else:
power_iteration_normalizer = "LU"
# Perform power iterations with Q to further 'imprint' the top
# singular vectors of A in Q
for i in range(n_iter):
if power_iteration_normalizer == "none":
Q = safe_sparse_dot(A, Q)
Q = safe_sparse_dot(A.T, Q)
elif power_iteration_normalizer == "LU":
Q, _ = linalg.lu(safe_sparse_dot(A, Q), permute_l=True)
Q, _ = linalg.lu(safe_sparse_dot(A.T, Q), permute_l=True)
elif power_iteration_normalizer == "QR":
Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode="economic")
Q, _ = linalg.qr(safe_sparse_dot(A.T, Q), mode="economic")
# Sample the range of A using by linear projection of Q
# Extract an orthonormal basis
Q, _ = linalg.qr(safe_sparse_dot(A, Q), mode="economic")
return Q
def randomized_svd(
M,
n_components,
*,
n_oversamples=10,
n_iter="auto",
power_iteration_normalizer="auto",
transpose="auto",
flip_sign=True,
random_state=None,
svd_lapack_driver="gesdd",
):
"""Compute a truncated randomized SVD.
This method solves the fixed-rank approximation problem described in [1]_
(problem (1.5), p5).
Parameters
----------
M : {ndarray, sparse matrix}
Matrix to decompose.
n_components : int
Number of singular values 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 singular vectors and singular values. 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 [1]_
(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 [1]_ page 9).
.. versionchanged:: 0.18
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.
.. versionadded:: 0.18
transpose : bool or 'auto', default='auto'
Whether the algorithm should be applied to M.T instead of M. The
result should approximately be the same. The 'auto' mode will
trigger the transposition if M.shape[1] > M.shape[0] since this
implementation of randomized SVD tend to be a little faster in that
case.
.. versionchanged:: 0.18
flip_sign : bool, default=True
The output of a singular value decomposition is only unique up to a
permutation of the signs of the singular vectors. If `flip_sign` is
set to `True`, the sign ambiguity is resolved by making the largest
loadings for each component in the left singular vectors positive.
random_state : int, RandomState instance or None, default='warn'
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>`.
.. versionchanged:: 1.2
The default value changed from 0 to None.
svd_lapack_driver : {"gesdd", "gesvd"}, default="gesdd"
Whether to use the more efficient divide-and-conquer approach
(`"gesdd"`) or more general rectangular approach (`"gesvd"`) to compute
the SVD of the matrix B, which is the projection of M into a low
dimensional subspace, as described in [1]_.
.. versionadded:: 1.2
Returns
-------
u : ndarray of shape (n_samples, n_components)
Unitary matrix having left singular vectors with signs flipped as columns.
s : ndarray of shape (n_components,)
The singular values, sorted in non-increasing order.
vh : ndarray of shape (n_components, n_features)
Unitary matrix having right singular vectors with signs flipped as rows.
Notes
-----
This algorithm finds a (usually very good) approximate truncated
singular value decomposition using randomization to speed up the
computations. It 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.
References
----------
.. [1] :arxiv:`"Finding structure with randomness:
Stochastic algorithms for constructing approximate matrix decompositions"
<0909.4061>`
Halko, et al. (2009)
.. [2] A randomized algorithm for the decomposition of matrices
Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
.. [3] An implementation of a randomized algorithm for principal component
analysis A. Szlam et al. 2014
Examples
--------
>>> import numpy as np
>>> from sklearn.utils.extmath import randomized_svd
>>> a = np.array([[1, 2, 3, 5],
... [3, 4, 5, 6],
... [7, 8, 9, 10]])
>>> U, s, Vh = randomized_svd(a, n_components=2, random_state=0)
>>> U.shape, s.shape, Vh.shape
((3, 2), (2,), (2, 4))
"""
if isinstance(M, (sparse.lil_matrix, sparse.dok_matrix)):
warnings.warn(
"Calculating SVD of a {} is expensive. "
"csr_matrix is more efficient.".format(type(M).__name__),
sparse.SparseEfficiencyWarning,
)
random_state = check_random_state(random_state)
n_random = n_components + n_oversamples
n_samples, n_features = M.shape
if n_iter == "auto":
# Checks if the number of iterations is explicitly specified
# 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
if transpose:
# this implementation is a bit faster with smaller shape[1]
M = M.T
Q = randomized_range_finder(
M,
size=n_random,
n_iter=n_iter,
power_iteration_normalizer=power_iteration_normalizer,
random_state=random_state,
)
# project M to the (k + p) dimensional space using the basis vectors
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)
del B
U = np.dot(Q, Uhat)
if flip_sign:
if not transpose:
U, Vt = svd_flip(U, Vt)
else:
# In case of transpose u_based_decision=false
# to actually flip based on u and not v.
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