"""Compute the log-likelihood of a rank ``rank`` dataset.

The dataset is assumed to be embedded in gaussian noise of shape(n,

dimf) having spectrum ``spectrum``. This implements the method of

T. P. Minka.

Parameters

----------

spectrum : ndarray of shape (n_features,)

Data spectrum.

rank : int

Tested rank value. It should be strictly lower than n_features,

otherwise the method isn't specified (division by zero in equation

(31) from the paper).

n_samples : int

Number of samples.

Returns

-------

ll : float

The log-likelihood.

References

----------

This implements the method of `Thomas P. Minka:

Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604

<https://proceedings.neurips.cc/paper/2000/file/7503cfacd12053d309b6bed5c89de212-Paper.pdf>`_

"""

xp, _ = get_namespace(spectrum)

n_features = spectrum.shape[0]

if not 1 <= rank < n_features:

raise ValueError("the tested rank should be in [1, n_features - 1]")

eps = 1e-15

if spectrum[rank - 1] < eps:

# When the tested rank is associated with a small eigenvalue, there's

# no point in computing the log-likelihood: it's going to be very

# small and won't be the max anyway. Also, it can lead to numerical

# issues below when computing pa, in particular in log((spectrum[i] -

# spectrum[j]) because this will take the log of something very small.

return -xp.inf

pu = -rank * log(2.0)

for i in range(1, rank + 1):

pu += (

gammaln((n_features - i + 1) / 2.0)

- log(xp.pi) * (n_features - i + 1) / 2.0

)

pl = xp.sum(xp.log(spectrum[:rank]))

pl = -pl * n_samples / 2.0

v = max(eps, xp.sum(spectrum[rank:]) / (n_features - rank))

pv = -log(v) * n_samples * (n_features - rank) / 2.0

m = n_features * rank - rank * (rank + 1.0) / 2.0

pp = log(2.0 * xp.pi) * (m + rank) / 2.0

pa = 0.0

spectrum_ = xp.asarray(spectrum, copy=True)

spectrum_[rank:n_features] = v

for i in range(rank):

for j in range(i + 1, spectrum.shape[0]):

pa += log(

(spectrum[i] - spectrum[j]) * (1.0 / spectrum_[j] - 1.0 / spectrum_[i])

) + log(n_samples)

ll = pu + pl + pv + pp - pa / 2.0 - rank * log(n_samples) / 2.0

"""Infers the dimension of a dataset with a given spectrum.

The returned value will be in [1, n_features - 1].

"""

xp, _ = get_namespace(spectrum)

ll = xp.empty_like(spectrum)

ll[0] = -xp.inf # we don't want to return n_components = 0

for rank in range(1, spectrum.shape[0]):

ll[rank] = _assess_dimension(spectrum, rank, n_samples)

"""Principal component analysis (PCA).

Linear dimensionality reduction using Singular Value Decomposition of the

data to project it to a lower dimensional space. The input data is centered

but not scaled for each feature before applying the SVD.

It uses the LAPACK implementation of the full SVD or a randomized truncated

SVD by the method of Halko et al. 2009, depending on the shape of the input

data and the number of components to extract.

With sparse inputs, the ARPACK implementation of the truncated SVD can be

used (i.e. through :func:`scipy.sparse.linalg.svds`). Alternatively, one

may consider :class:`TruncatedSVD` where the data are not centered.

Notice that this class only supports sparse inputs for some solvers such as

"arpack" and "covariance_eigh". See :class:`TruncatedSVD` for an

alternative with sparse data.

For a usage example, see

:ref:`sphx_glr_auto_examples_decomposition_plot_pca_iris.py`

Read more in the :ref:`User Guide <PCA>`.

Parameters

----------

n_components : int, float or 'mle', default=None

Number of components to keep.

if n_components is not set all components are kept::

n_components == min(n_samples, n_features)

If ``n_components == 'mle'`` and ``svd_solver == 'full'``, Minka's

MLE is used to guess the dimension. Use of ``n_components == 'mle'``

will interpret ``svd_solver == 'auto'`` as ``svd_solver == 'full'``.

If ``0 < n_components < 1`` and ``svd_solver == 'full'``, select the

number of components such that the amount of variance that needs to be

explained is greater than the percentage specified by n_components.

If ``svd_solver == 'arpack'``, the number of components must be

strictly less than the minimum of n_features and n_samples.

Hence, the None case results in::

n_components == min(n_samples, n_features) - 1

copy : bool, default=True

If False, data passed to fit are overwritten and running

fit(X).transform(X) will not yield the expected results,

use fit_transform(X) instead.

whiten : bool, default=False

When True (False by default) the `components_` vectors are multiplied

by the square root of n_samples and then divided by the singular values

to ensure uncorrelated outputs with unit component-wise variances.

Whitening will remove some information from the transformed signal

(the relative variance scales of the components) but can sometime

improve the predictive accuracy of the downstream estimators by

making their data respect some hard-wired assumptions.

svd_solver : {'auto', 'full', 'covariance_eigh', 'arpack', 'randomized'},\

default='auto'

"auto" :

The solver is selected by a default 'auto' policy is based on `X.shape` and

`n_components`: if the input data has fewer than 1000 features and

more than 10 times as many samples, then the "covariance_eigh"

solver is used. Otherwise, if the input data is larger than 500x500

and the number of components to extract is lower than 80% of the

smallest dimension of the data, then the more efficient

"randomized" method is selected. Otherwise the exact "full" SVD is

computed and optionally truncated afterwards.

"full" :

Run exact full SVD calling the standard LAPACK solver via

`scipy.linalg.svd` and select the components by postprocessing

"covariance_eigh" :

Precompute the covariance matrix (on centered data), run a

classical eigenvalue decomposition on the covariance matrix

typically using LAPACK and select the components by postprocessing.

This solver is very efficient for n_samples >> n_features and small

n_features. It is, however, not tractable otherwise for large

n_features (large memory footprint required to materialize the

covariance matrix). Also note that compared to the "full" solver,

this solver effectively doubles the condition number and is

therefore less numerical stable (e.g. on input data with a large

range of singular values).

"arpack" :

Run SVD truncated to `n_components` calling ARPACK solver via

`scipy.sparse.linalg.svds`. It requires strictly

`0 < n_components < min(X.shape)`

"randomized" :

Run randomized SVD by the method of Halko et al.

.. versionadded:: 0.18.0

.. versionchanged:: 1.5

Added the 'covariance_eigh' solver.

tol : float, default=0.0

Tolerance for singular values computed by svd_solver == 'arpack'.

Must be of range [0.0, infinity).

.. versionadded:: 0.18.0

iterated_power : int or 'auto', default='auto'

Number of iterations for the power method computed by

svd_solver == 'randomized'.

Must be of range [0, infinity).

.. versionadded:: 0.18.0

n_oversamples : int, default=10

This parameter is only relevant when `svd_solver="randomized"`.

It corresponds to the additional number of random vectors to sample the

range of `X` so as to ensure proper conditioning. See

:func:`~sklearn.utils.extmath.randomized_svd` for more details.

.. versionadded:: 1.1

power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'

Power iteration normalizer for randomized SVD solver.

Not used by ARPACK. See :func:`~sklearn.utils.extmath.randomized_svd`

for more details.

.. versionadded:: 1.1

random_state : int, RandomState instance or None, default=None

Used when the 'arpack' or 'randomized' solvers are used. Pass an int

for reproducible results across multiple function calls.

See :term:`Glossary <random_state>`.

.. versionadded:: 0.18.0

Attributes

----------

components_ : ndarray of shape (n_components, n_features)

Principal axes in feature space, representing the directions of

maximum variance in the data. Equivalently, the right singular

vectors of the centered input data, parallel to its eigenvectors.

The components are sorted by decreasing ``explained_variance_``.

explained_variance_ : ndarray of shape (n_components,)

The amount of variance explained by each of the selected components.

The variance estimation uses `n_samples - 1` degrees of freedom.

Equal to n_components largest eigenvalues

of the covariance matrix of X.

.. versionadded:: 0.18

explained_variance_ratio_ : ndarray of shape (n_components,)

Percentage of variance explained by each of the selected components.

If ``n_components`` is not set then all components are stored and the

sum of the ratios is equal to 1.0.

singular_values_ : ndarray of shape (n_components,)

The singular values corresponding to each of the selected components.

The singular values are equal to the 2-norms of the ``n_components``

variables in the lower-dimensional space.

.. versionadded:: 0.19

mean_ : ndarray of shape (n_features,)

Per-feature empirical mean, estimated from the training set.

Equal to `X.mean(axis=0)`.

n_components_ : int

The estimated number of components. When n_components is set

to 'mle' or a number between 0 and 1 (with svd_solver == 'full') this

number is estimated from input data. Otherwise it equals the parameter

n_components, or the lesser value of n_features and n_samples

if n_components is None.

n_samples_ : int

Number of samples in the training data.

noise_variance_ : float

The estimated noise covariance following the Probabilistic PCA model

from Tipping and Bishop 1999. See "Pattern Recognition and

Machine Learning" by C. Bishop, 12.2.1 p. 574 or

http://www.miketipping.com/papers/met-mppca.pdf. It is required to

compute the estimated data covariance and score samples.

Equal to the average of (min(n_features, n_samples) - n_components)

smallest eigenvalues of the covariance matrix of X.

n_features_in_ : int

Number of features seen during :term:`fit`.

.. versionadded:: 0.24

feature_names_in_ : ndarray of shape (`n_features_in_`,)

Names of features seen during :term:`fit`. Defined only when `X`

has feature names that are all strings.

.. versionadded:: 1.0

See Also

--------

KernelPCA : Kernel Principal Component Analysis.

SparsePCA : Sparse Principal Component Analysis.

TruncatedSVD : Dimensionality reduction using truncated SVD.

IncrementalPCA : Incremental Principal Component Analysis.

References

----------

For n_components == 'mle', this class uses the method from:

`Minka, T. P.. "Automatic choice of dimensionality for PCA".

In NIPS, pp. 598-604 <https://tminka.github.io/papers/pca/minka-pca.pdf>`_

Implements the probabilistic PCA model from:

`Tipping, M. E., and Bishop, C. M. (1999). "Probabilistic principal

component analysis". Journal of the Royal Statistical Society:

Series B (Statistical Methodology), 61(3), 611-622.

<http://www.miketipping.com/papers/met-mppca.pdf>`_

via the score and score_samples methods.

For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`.

For svd_solver == 'randomized', see:

:doi:`Halko, N., Martinsson, P. G., and Tropp, J. A. (2011).

"Finding structure with randomness: Probabilistic algorithms for

constructing approximate matrix decompositions".

SIAM review, 53(2), 217-288.

<10.1137/090771806>`

and also

:doi:`Martinsson, P. G., Rokhlin, V., and Tygert, M. (2011).

"A randomized algorithm for the decomposition of matrices".

Applied and Computational Harmonic Analysis, 30(1), 47-68.

<10.1016/j.acha.2010.02.003>`

Examples

--------

>>> import numpy as np

>>> from sklearn.decomposition import PCA

>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])

>>> pca = PCA(n_components=2)

>>> pca.fit(X)

PCA(n_components=2)

>>> print(pca.explained_variance_ratio_)

[0.9924... 0.0075...]

>>> print(pca.singular_values_)

[6.30061... 0.54980...]

>>> pca = PCA(n_components=2, svd_solver='full')

>>> pca.fit(X)

PCA(n_components=2, svd_solver='full')

>>> print(pca.explained_variance_ratio_)

[0.9924... 0.00755...]

>>> print(pca.singular_values_)

[6.30061... 0.54980...]

>>> pca = PCA(n_components=1, svd_solver='arpack')

>>> pca.fit(X)

PCA(n_components=1, svd_solver='arpack')

>>> print(pca.explained_variance_ratio_)

[0.99244...]

>>> print(pca.singular_values_)

[6.30061...]

"""

_parameter_constraints: dict = {

"n_components": [

Interval(Integral, 0, None, closed="left"),

Interval(RealNotInt, 0, 1, closed="neither"),

StrOptions({"mle"}),

None,

],

"copy": ["boolean"],

"whiten": ["boolean"],

"svd_solver": [

StrOptions({"auto", "full", "covariance_eigh", "arpack", "randomized"})

],

"tol": [Interval(Real, 0, None, closed="left")],

"iterated_power": [

StrOptions({"auto"}),

Interval(Integral, 0, None, closed="left"),

],

"n_oversamples": [Interval(Integral, 1, None, closed="left")],

"power_iteration_normalizer": [StrOptions({"auto", "QR", "LU", "none"})],

"random_state": ["random_state"],

}

def __init__(

self,

n_components=None,

*,

copy=True,

whiten=False,

svd_solver="auto",

tol=0.0,

iterated_power="auto",

n_oversamples=10,

power_iteration_normalizer="auto",

random_state=None,

):

self.n_components = n_components

self.copy = copy

self.whiten = whiten

self.svd_solver = svd_solver

self.tol = tol

self.iterated_power = iterated_power

self.n_oversamples = n_oversamples

self.power_iteration_normalizer = power_iteration_normalizer

self.random_state = random_state

@_fit_context(prefer_skip_nested_validation=True)

def fit(self, X, y=None):

"""Fit the model with X.

Parameters

----------

X : {array-like, sparse matrix} of shape (n_samples, n_features)

Training data, where `n_samples` is the number of samples

and `n_features` is the number of features.

y : Ignored

Ignored.

Returns

-------

self : object

Returns the instance itself.

"""

self._fit(X)

return self

@_fit_context(prefer_skip_nested_validation=True)

def fit_transform(self, X, y=None):

"""Fit the model with X and apply the dimensionality reduction on X.

Parameters

----------

X : {array-like, sparse matrix} of shape (n_samples, n_features)

Training data, where `n_samples` is the number of samples

and `n_features` is the number of features.

y : Ignored

Ignored.

Returns

-------

X_new : ndarray of shape (n_samples, n_components)

Transformed values.

Notes

-----

This method returns a Fortran-ordered array. To convert it to a

C-ordered array, use 'np.ascontiguousarray'.

"""

U, S, _, X, x_is_centered, xp = self._fit(X)

if U is not None:

U = U[:, : self.n_components_]

if self.whiten:

# X_new = X * V / S * sqrt(n_samples) = U * sqrt(n_samples)

U *= sqrt(X.shape[0] - 1)

else:

# X_new = X * V = U * S * Vt * V = U * S

U *= S[: self.n_components_]

return U

else: # solver="covariance_eigh" does not compute U at fit time.

return self._transform(X, xp, x_is_centered=x_is_centered)

def _fit(self, X):

"""Dispatch to the right submethod depending on the chosen solver."""

xp, is_array_api_compliant = get_namespace(X)

# Raise an error for sparse input and unsupported svd_solver

if issparse(X) and self.svd_solver not in ["auto", "arpack", "covariance_eigh"]:

raise TypeError(

'PCA only support sparse inputs with the "arpack" and'

f' "covariance_eigh" solvers, while "{self.svd_solver}" was passed. See'

" TruncatedSVD for a possible alternative."

)

if self.svd_solver == "arpack" and is_array_api_compliant:

raise ValueError(

"PCA with svd_solver='arpack' is not supported for Array API inputs."

)

# Validate the data, without ever forcing a copy as any solver that

# supports sparse input data and the `covariance_eigh` solver are

# written in a way to avoid the need for any inplace modification of

# the input data contrary to the other solvers.

# The copy will happen

# later, only if needed, once the solver negotiation below is done.

X = validate_data(

self,

X,

dtype=[xp.float64, xp.float32],

force_writeable=True,

accept_sparse=("csr", "csc"),

ensure_2d=True,

copy=False,

)

self._fit_svd_solver = self.svd_solver

if self._fit_svd_solver == "auto" and issparse(X):

self._fit_svd_solver = "arpack"

if self.n_components is None:

if self._fit_svd_solver != "arpack":

n_components = min(X.shape)

else:

n_components = min(X.shape) - 1

else:

n_components = self.n_components

if self._fit_svd_solver == "auto":

# Tall and skinny problems are best handled by precomputing the

# covariance matrix.

if X.shape[1] <= 1_000 and X.shape[0] >= 10 * X.shape[1]:

self._fit_svd_solver = "covariance_eigh"

# Small problem or n_components == 'mle', just call full PCA

elif max(X.shape) <= 500 or n_components == "mle":

self._fit_svd_solver = "full"

elif 1 <= n_components < 0.8 * min(X.shape):

self._fit_svd_solver = "randomized"

# This is also the case of n_components in (0, 1)

else:

self._fit_svd_solver = "full"

# Call different fits for either full or truncated SVD

if self._fit_svd_solver in ("full", "covariance_eigh"):

return self._fit_full(X, n_components, xp, is_array_api_compliant)

elif self._fit_svd_solver in ["arpack", "randomized"]:

return self._fit_truncated(X, n_components, xp)

def _fit_full(self, X, n_components, xp, is_array_api_compliant):

"""Fit the model by computing full SVD on X."""

n_samples, n_features = X.shape

if n_components == "mle":

if n_samples < n_features:

raise ValueError(

"n_components='mle' is only supported if n_samples >= n_features"

)

elif not 0 <= n_components <= min(n_samples, n_features):

raise ValueError(

f"n_components={n_components} must be between 0 and "

f"min(n_samples, n_features)={min(n_samples, n_features)} with "

f"svd_solver={self._fit_svd_solver!r}"

)

self.mean_ = xp.mean(X, axis=0)

# When X is a scipy sparse matrix, self.mean_ is a numpy matrix, so we need

# to transform it to a 1D array. Note that this is not the case when X

# is a scipy sparse array.

# TODO: remove the following two lines when scikit-learn only depends

# on scipy versions that no longer support scipy.sparse matrices.

self.mean_ = xp.reshape(xp.asarray(self.mean_), (-1,))

if self._fit_svd_solver == "full":

X_centered = xp.asarray(X, copy=True) if self.copy else X

X_centered -= self.mean_

x_is_centered = not self.copy

if not is_array_api_compliant:

# Use scipy.linalg with NumPy/SciPy inputs for the sake of not

# introducing unanticipated behavior changes. In the long run we

# could instead decide to always use xp.linalg.svd for all inputs,

# but that would make this code rely on numpy's SVD instead of

# scipy's. It's not 100% clear whether they use the same LAPACK

# solver by default though (assuming both are built against the

# same BLAS).

U, S, Vt = linalg.svd(X_centered, full_matrices=False)

else:

U, S, Vt = xp.linalg.svd(X_centered, full_matrices=False)

explained_variance_ = (S**2) / (n_samples - 1)

else:

assert self._fit_svd_solver == "covariance_eigh"

# In the following, we center the covariance matrix C afterwards

# (without centering the data X first) to avoid an unnecessary copy

# of X. Note that the mean_ attribute is still needed to center

# test data in the transform method.

#

# Note: at the time of writing, `xp.cov` does not exist in the

# Array API standard:

# https://github.com/data-apis/array-api/issues/43

#

# Besides, using `numpy.cov`, as of numpy 1.26.0, would not be

# memory efficient for our use case when `n_samples >> n_features`:

# `numpy.cov` centers a copy of the data before computing the

# matrix product instead of subtracting a small `(n_features,

# n_features)` square matrix from the gram matrix X.T @ X, as we do

# below.

x_is_centered = False

C = X.T @ X

C -= (

n_samples

* xp.reshape(self.mean_, (-1, 1))

* xp.reshape(self.mean_, (1, -1))

)

C /= n_samples - 1

eigenvals, eigenvecs = xp.linalg.eigh(C)

# When X is a scipy sparse matrix, the following two datastructures

# are returned as instances of the soft-deprecated numpy.matrix

# class. Note that this problem does not occur when X is a scipy

# sparse array (or another other kind of supported array).

# TODO: remove the following two lines when scikit-learn only

# depends on scipy versions that no longer support scipy.sparse

# matrices.

eigenvals = xp.reshape(xp.asarray(eigenvals), (-1,))

eigenvecs = xp.asarray(eigenvecs)

eigenvals = xp.flip(eigenvals, axis=0)

eigenvecs = xp.flip(eigenvecs, axis=1)

# The covariance matrix C is positive semi-definite by

# construction. However, the eigenvalues returned by xp.linalg.eigh

# can be slightly negative due to numerical errors. This would be

# an issue for the subsequent sqrt, hence the manual clipping.

eigenvals[eigenvals < 0.0] = 0.0

explained_variance_ = eigenvals

# Re-construct SVD of centered X indirectly and make it consistent

# with the other solvers.

S = xp.sqrt(eigenvals * (n_samples - 1))

Vt = eigenvecs.T

U = None

# flip eigenvectors' sign to enforce deterministic output

U, Vt = svd_flip(U, Vt, u_based_decision=False)

components_ = Vt

# Get variance explained by singular values

total_var = xp.sum(explained_variance_)

explained_variance_ratio_ = explained_variance_ / total_var

singular_values_ = xp.asarray(S, copy=True) # Store the singular values.

# Postprocess the number of components required

if n_components == "mle":

n_components = _infer_dimension(explained_variance_, n_samples)

elif 0 < n_components < 1.0:

# number of components for which the cumulated explained

# variance percentage is superior to the desired threshold

# side='right' ensures that number of features selected

# their variance is always greater than n_components float

# passed. More discussion in issue: #15669

if is_array_api_compliant:

# Convert to numpy as xp.cumsum and xp.searchsorted are not

# part of the Array API standard yet:

#

# https://github.com/data-apis/array-api/issues/597

# https://github.com/data-apis/array-api/issues/688

#

# Furthermore, it's not always safe to call them for namespaces

# that already implement them: for instance as

# cupy.searchsorted does not accept a float as second argument.

explained_variance_ratio_np = _convert_to_numpy(

explained_variance_ratio_, xp=xp

)

else:

explained_variance_ratio_np = explained_variance_ratio_

ratio_cumsum = stable_cumsum(explained_variance_ratio_np)

n_components = np.searchsorted(ratio_cumsum, n_components, side="right") + 1

# Compute noise covariance using Probabilistic PCA model

# The sigma2 maximum likelihood (cf. eq. 12.46)

if n_components < min(n_features, n_samples):

self.noise_variance_ = xp.mean(explained_variance_[n_components:])

else:

self.noise_variance_ = 0.0

self.n_samples_ = n_samples

self.n_components_ = n_components

# Assign a copy of the result of the truncation of the components in

# order to:

# - release the memory used by the discarded components,

# - ensure that the kept components are allocated contiguously in

# memory to make the transform method faster by leveraging cache

# locality.

self.components_ = xp.asarray(components_[:n_components, :], copy=True)

# We do the same for the other arrays for the sake of consistency.

self.explained_variance_ = xp.asarray(

explained_variance_[:n_components], copy=True

)

self.explained_variance_ratio_ = xp.asarray(

explained_variance_ratio_[:n_components], copy=True

)

self.singular_values_ = xp.asarray(singular_values_[:n_components], copy=True)

return U, S, Vt, X, x_is_centered, xp

def _fit_truncated(self, X, n_components, xp):

"""Fit the model by computing truncated SVD (by ARPACK or randomized)

on X.

"""

n_samples, n_features = X.shape

svd_solver = self._fit_svd_solver

if isinstance(n_components, str):

raise ValueError(

"n_components=%r cannot be a string with svd_solver='%s'"

% (n_components, svd_solver)

)

elif not 1 <= n_components <= min(n_samples, n_features):

raise ValueError(

"n_components=%r must be between 1 and "

"min(n_samples, n_features)=%r with "

"svd_solver='%s'"

% (n_components, min(n_samples, n_features), svd_solver)

)

elif svd_solver == "arpack" and n_components == min(n_samples, n_features):

raise ValueError(

"n_components=%r must be strictly less than "

"min(n_samples, n_features)=%r with "

"svd_solver='%s'"

% (n_components, min(n_samples, n_features), svd_solver)

)

random_state = check_random_state(self.random_state)

# Center data

total_var = None

if issparse(X):

self.mean_, var = mean_variance_axis(X, axis=0)

total_var = var.sum() * n_samples / (n_samples - 1) # ddof=1

X_centered = _implicit_column_offset(X, self.mean_)

x_is_centered = False

else:

self.mean_ = xp.mean(X, axis=0)

X_centered = xp.asarray(X, copy=True) if self.copy else X

X_centered -= self.mean_

x_is_centered = not self.copy

if svd_solver == "arpack":

v0 = _init_arpack_v0(min(X.shape), random_state)

U, S, Vt = svds(X_centered, k=n_components, tol=self.tol, v0=v0)

# svds doesn't abide by scipy.linalg.svd/randomized_svd

# conventions, so reverse its outputs.

S = S[::-1]

# flip eigenvectors' sign to enforce deterministic output

U, Vt = svd_flip(U[:, ::-1], Vt[::-1], u_based_decision=False)

elif svd_solver == "randomized":

# sign flipping is done inside

U, S, Vt = randomized_svd(

X_centered,

n_components=n_components,

n_oversamples=self.n_oversamples,

n_iter=self.iterated_power,

power_iteration_normalizer=self.power_iteration_normalizer,

flip_sign=False,

random_state=random_state,

)

U, Vt = svd_flip(U, Vt, u_based_decision=False)

self.n_samples_ = n_samples

self.components_ = Vt

self.n_components_ = n_components

# Get variance explained by singular values

self.explained_variance_ = (S**2) / (n_samples - 1)

# Workaround in-place variance calculation since at the time numpy

# did not have a way to calculate variance in-place.

#

# TODO: update this code to either:

# * Use the array-api variance calculation, unless memory usage suffers

# * Update sklearn.utils.extmath._incremental_mean_and_var to support array-api

# See: https://github.com/scikit-learn/scikit-learn/pull/18689#discussion_r1335540991

if total_var is None:

N = X.shape[0] - 1

X_centered **= 2

total_var = xp.sum(X_centered) / N

self.explained_variance_ratio_ = self.explained_variance_ / total_var

self.singular_values_ = xp.asarray(S, copy=True) # Store the singular values.

if self.n_components_ < min(n_features, n_samples):

self.noise_variance_ = total_var - xp.sum(self.explained_variance_)

self.noise_variance_ /= min(n_features, n_samples) - n_components

else:

self.noise_variance_ = 0.0

return U, S, Vt, X, x_is_centered, xp

def score_samples(self, X):

"""Return the log-likelihood of each sample.

See. "Pattern Recognition and Machine Learning"

by C. Bishop, 12.2.1 p. 574

or http://www.miketipping.com/papers/met-mppca.pdf

Parameters

----------

X : array-like of shape (n_samples, n_features)

The data.

Returns

-------

ll : ndarray of shape (n_samples,)

Log-likelihood of each sample under the current model.

"""

check_is_fitted(self)

xp, _ = get_namespace(X)

X = validate_data(self, X, dtype=[xp.float64, xp.float32], reset=False)

Xr = X - self.mean_

n_features = X.shape[1]

precision = self.get_precision()

log_like = -0.5 * xp.sum(Xr * (Xr @ precision), axis=1)

log_like -= 0.5 * (n_features * log(2.0 * np.pi) - fast_logdet(precision))

return log_like

def score(self, X, y=None):

"""Return the average log-likelihood of all samples.

See. "Pattern Recognition and Machine Learning"

by C. Bishop, 12.2.1 p. 574

or http://www.miketipping.com/papers/met-mppca.pdf

Parameters

----------

X : array-like of shape (n_samples, n_features)

The data.

y : Ignored

Ignored.

Returns

-------

ll : float

Average log-likelihood of the samples under the current model.

"""

xp, _ = get_namespace(X)

return float(xp.mean(self.score_samples(X)))

def __sklearn_tags__(self):

tags = super().__sklearn_tags__()

tags.transformer_tags.preserves_dtype = ["float64", "float32"]

tags.array_api_support = True

tags.input_tags.sparse = self.svd_solver in (

"auto",

"arpack",

"covariance_eigh",

)