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"""Approximate kernel feature maps based on Fourier transforms and count sketches."""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import warnings
from numbers import Integral, Real
import numpy as np
import scipy.sparse as sp
from scipy.fft import fft, ifft
from scipy.linalg import svd
from .base import (
BaseEstimator,
ClassNamePrefixFeaturesOutMixin,
TransformerMixin,
_fit_context,
)
from .metrics.pairwise import KERNEL_PARAMS, PAIRWISE_KERNEL_FUNCTIONS, pairwise_kernels
from .utils import check_random_state
from .utils._param_validation import Interval, StrOptions
from .utils.extmath import safe_sparse_dot
from .utils.validation import (
_check_feature_names_in,
check_is_fitted,
validate_data,
)
class PolynomialCountSketch(
ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator
):
"""Polynomial kernel approximation via Tensor Sketch.
Implements Tensor Sketch, which approximates the feature map
of the polynomial kernel::
K(X, Y) = (gamma * <X, Y> + coef0)^degree
by efficiently computing a Count Sketch of the outer product of a
vector with itself using Fast Fourier Transforms (FFT). Read more in the
:ref:`User Guide <polynomial_kernel_approx>`.
.. versionadded:: 0.24
Parameters
----------
gamma : float, default=1.0
Parameter of the polynomial kernel whose feature map
will be approximated.
degree : int, default=2
Degree of the polynomial kernel whose feature map
will be approximated.
coef0 : int, default=0
Constant term of the polynomial kernel whose feature map
will be approximated.
n_components : int, default=100
Dimensionality of the output feature space. Usually, `n_components`
should be greater than the number of features in input samples in
order to achieve good performance. The optimal score / run time
balance is typically achieved around `n_components` = 10 * `n_features`,
but this depends on the specific dataset being used.
random_state : int, RandomState instance, default=None
Determines random number generation for indexHash and bitHash
initialization. Pass an int for reproducible results across multiple
function calls. See :term:`Glossary <random_state>`.
Attributes
----------
indexHash_ : ndarray of shape (degree, n_features), dtype=int64
Array of indexes in range [0, n_components) used to represent
the 2-wise independent hash functions for Count Sketch computation.
bitHash_ : ndarray of shape (degree, n_features), dtype=float32
Array with random entries in {+1, -1}, used to represent
the 2-wise independent hash functions for Count Sketch computation.
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
--------
AdditiveChi2Sampler : Approximate feature map for additive chi2 kernel.
Nystroem : Approximate a kernel map using a subset of the training data.
RBFSampler : Approximate a RBF kernel feature map using random Fourier
features.
SkewedChi2Sampler : Approximate feature map for "skewed chi-squared" kernel.
sklearn.metrics.pairwise.kernel_metrics : List of built-in kernels.
Examples
--------
>>> from sklearn.kernel_approximation import PolynomialCountSketch
>>> from sklearn.linear_model import SGDClassifier
>>> X = [[0, 0], [1, 1], [1, 0], [0, 1]]
>>> y = [0, 0, 1, 1]
>>> ps = PolynomialCountSketch(degree=3, random_state=1)
>>> X_features = ps.fit_transform(X)
>>> clf = SGDClassifier(max_iter=10, tol=1e-3)
>>> clf.fit(X_features, y)
SGDClassifier(max_iter=10)
>>> clf.score(X_features, y)
1.0
For a more detailed example of usage, see
:ref:`sphx_glr_auto_examples_kernel_approximation_plot_scalable_poly_kernels.py`
"""
_parameter_constraints: dict = {
"gamma": [Interval(Real, 0, None, closed="left")],
"degree": [Interval(Integral, 1, None, closed="left")],
"coef0": [Interval(Real, None, None, closed="neither")],
"n_components": [Interval(Integral, 1, None, closed="left")],
"random_state": ["random_state"],
}
def __init__(
self, *, gamma=1.0, degree=2, coef0=0, n_components=100, random_state=None
):
self.gamma = gamma
self.degree = degree
self.coef0 = coef0
self.n_components = n_components
self.random_state = random_state
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit the model with X.
Initializes the internal variables. The method needs no information
about the distribution of data, so we only care about n_features in 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 : array-like of shape (n_samples,) or (n_samples, n_outputs), \
default=None
Target values (None for unsupervised transformations).
Returns
-------
self : object
Returns the instance itself.
"""
X = validate_data(self, X, accept_sparse="csc")
random_state = check_random_state(self.random_state)
n_features = X.shape[1]
if self.coef0 != 0:
n_features += 1
self.indexHash_ = random_state.randint(
0, high=self.n_components, size=(self.degree, n_features)
)
self.bitHash_ = random_state.choice(a=[-1, 1], size=(self.degree, n_features))
self._n_features_out = self.n_components
return self
def transform(self, X):
"""Generate the feature map approximation for X.
Parameters
----------
X : {array-like}, shape (n_samples, n_features)
New data, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
Returns the instance itself.
"""
check_is_fitted(self)
X = validate_data(self, X, accept_sparse="csc", reset=False)
X_gamma = np.sqrt(self.gamma) * X
if sp.issparse(X_gamma) and self.coef0 != 0:
X_gamma = sp.hstack(
[X_gamma, np.sqrt(self.coef0) * np.ones((X_gamma.shape[0], 1))],
format="csc",
)
elif not sp.issparse(X_gamma) and self.coef0 != 0:
X_gamma = np.hstack(
[X_gamma, np.sqrt(self.coef0) * np.ones((X_gamma.shape[0], 1))]
)
if X_gamma.shape[1] != self.indexHash_.shape[1]:
raise ValueError(
"Number of features of test samples does not"
" match that of training samples."
)
count_sketches = np.zeros((X_gamma.shape[0], self.degree, self.n_components))
if sp.issparse(X_gamma):
for j in range(X_gamma.shape[1]):
for d in range(self.degree):
iHashIndex = self.indexHash_[d, j]
iHashBit = self.bitHash_[d, j]
count_sketches[:, d, iHashIndex] += (
(iHashBit * X_gamma[:, [j]]).toarray().ravel()
)
else:
for j in range(X_gamma.shape[1]):
for d in range(self.degree):
iHashIndex = self.indexHash_[d, j]
iHashBit = self.bitHash_[d, j]
count_sketches[:, d, iHashIndex] += iHashBit * X_gamma[:, j]
# For each same, compute a count sketch of phi(x) using the polynomial
# multiplication (via FFT) of p count sketches of x.
count_sketches_fft = fft(count_sketches, axis=2, overwrite_x=True)
count_sketches_fft_prod = np.prod(count_sketches_fft, axis=1)
data_sketch = np.real(ifft(count_sketches_fft_prod, overwrite_x=True))
return data_sketch
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.input_tags.sparse = True
return tags
class RBFSampler(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
"""Approximate a RBF kernel feature map using random Fourier features.
It implements a variant of Random Kitchen Sinks.[1]
Read more in the :ref:`User Guide <rbf_kernel_approx>`.
Parameters
----------
gamma : 'scale' or float, default=1.0
Parameter of RBF kernel: exp(-gamma * x^2).
If ``gamma='scale'`` is passed then it uses
1 / (n_features * X.var()) as value of gamma.
.. versionadded:: 1.2
The option `"scale"` was added in 1.2.
n_components : int, default=100
Number of Monte Carlo samples per original feature.
Equals the dimensionality of the computed feature space.
random_state : int, RandomState instance or None, default=None
Pseudo-random number generator to control the generation of the random
weights and random offset when fitting the training data.
Pass an int for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Attributes
----------
random_offset_ : ndarray of shape (n_components,), dtype={np.float64, np.float32}
Random offset used to compute the projection in the `n_components`
dimensions of the feature space.
random_weights_ : ndarray of shape (n_features, n_components),\
dtype={np.float64, np.float32}
Random projection directions drawn from the Fourier transform
of the RBF kernel.
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
--------
AdditiveChi2Sampler : Approximate feature map for additive chi2 kernel.
Nystroem : Approximate a kernel map using a subset of the training data.
PolynomialCountSketch : Polynomial kernel approximation via Tensor Sketch.
SkewedChi2Sampler : Approximate feature map for
"skewed chi-squared" kernel.
sklearn.metrics.pairwise.kernel_metrics : List of built-in kernels.
Notes
-----
See "Random Features for Large-Scale Kernel Machines" by A. Rahimi and
Benjamin Recht.
[1] "Weighted Sums of Random Kitchen Sinks: Replacing
minimization with randomization in learning" by A. Rahimi and
Benjamin Recht.
(https://people.eecs.berkeley.edu/~brecht/papers/08.rah.rec.nips.pdf)
Examples
--------
>>> from sklearn.kernel_approximation import RBFSampler
>>> from sklearn.linear_model import SGDClassifier
>>> X = [[0, 0], [1, 1], [1, 0], [0, 1]]
>>> y = [0, 0, 1, 1]
>>> rbf_feature = RBFSampler(gamma=1, random_state=1)
>>> X_features = rbf_feature.fit_transform(X)
>>> clf = SGDClassifier(max_iter=5, tol=1e-3)
>>> clf.fit(X_features, y)
SGDClassifier(max_iter=5)
>>> clf.score(X_features, y)
1.0
"""
_parameter_constraints: dict = {
"gamma": [
StrOptions({"scale"}),
Interval(Real, 0.0, None, closed="left"),
],
"n_components": [Interval(Integral, 1, None, closed="left")],
"random_state": ["random_state"],
}
def __init__(self, *, gamma=1.0, n_components=100, random_state=None):
self.gamma = gamma
self.n_components = n_components
self.random_state = random_state
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit the model with X.
Samples random projection according to n_features.
Parameters
----------
X : {array-like, sparse matrix}, shape (n_samples, n_features)
Training data, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : array-like, shape (n_samples,) or (n_samples, n_outputs), \
default=None
Target values (None for unsupervised transformations).
Returns
-------
self : object
Returns the instance itself.
"""
X = validate_data(self, X, accept_sparse="csr")
random_state = check_random_state(self.random_state)
n_features = X.shape[1]
sparse = sp.issparse(X)
if self.gamma == "scale":
# var = E[X^2] - E[X]^2 if sparse
X_var = (X.multiply(X)).mean() - (X.mean()) ** 2 if sparse else X.var()
self._gamma = 1.0 / (n_features * X_var) if X_var != 0 else 1.0
else:
self._gamma = self.gamma
self.random_weights_ = (2.0 * self._gamma) ** 0.5 * random_state.normal(
size=(n_features, self.n_components)
)
self.random_offset_ = random_state.uniform(0, 2 * np.pi, size=self.n_components)
if X.dtype == np.float32:
# Setting the data type of the fitted attribute will ensure the
# output data type during `transform`.
self.random_weights_ = self.random_weights_.astype(X.dtype, copy=False)
self.random_offset_ = self.random_offset_.astype(X.dtype, copy=False)
self._n_features_out = self.n_components
return self
def transform(self, X):
"""Apply the approximate feature map to X.
Parameters
----------
X : {array-like, sparse matrix}, shape (n_samples, n_features)
New data, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
Returns the instance itself.
"""
check_is_fitted(self)
X = validate_data(self, X, accept_sparse="csr", reset=False)
projection = safe_sparse_dot(X, self.random_weights_)
projection += self.random_offset_
np.cos(projection, projection)
projection *= (2.0 / self.n_components) ** 0.5
return projection
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.input_tags.sparse = True
tags.transformer_tags.preserves_dtype = ["float64", "float32"]
return tags
class SkewedChi2Sampler(
ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator
):
"""Approximate feature map for "skewed chi-squared" kernel.
Read more in the :ref:`User Guide <skewed_chi_kernel_approx>`.
Parameters
----------
skewedness : float, default=1.0
"skewedness" parameter of the kernel. Needs to be cross-validated.
n_components : int, default=100
Number of Monte Carlo samples per original feature.
Equals the dimensionality of the computed feature space.
random_state : int, RandomState instance or None, default=None
Pseudo-random number generator to control the generation of the random
weights and random offset when fitting the training data.
Pass an int for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
Attributes
----------
random_weights_ : ndarray of shape (n_features, n_components)
Weight array, sampled from a secant hyperbolic distribution, which will
be used to linearly transform the log of the data.
random_offset_ : ndarray of shape (n_features, n_components)
Bias term, which will be added to the data. It is uniformly distributed
between 0 and 2*pi.
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
--------
AdditiveChi2Sampler : Approximate feature map for additive chi2 kernel.
Nystroem : Approximate a kernel map using a subset of the training data.
RBFSampler : Approximate a RBF kernel feature map using random Fourier
features.
SkewedChi2Sampler : Approximate feature map for "skewed chi-squared" kernel.
sklearn.metrics.pairwise.chi2_kernel : The exact chi squared kernel.
sklearn.metrics.pairwise.kernel_metrics : List of built-in kernels.
References
----------
See "Random Fourier Approximations for Skewed Multiplicative Histogram
Kernels" by Fuxin Li, Catalin Ionescu and Cristian Sminchisescu.
Examples
--------
>>> from sklearn.kernel_approximation import SkewedChi2Sampler
>>> from sklearn.linear_model import SGDClassifier
>>> X = [[0, 0], [1, 1], [1, 0], [0, 1]]
>>> y = [0, 0, 1, 1]
>>> chi2_feature = SkewedChi2Sampler(skewedness=.01,
... n_components=10,
... random_state=0)
>>> X_features = chi2_feature.fit_transform(X, y)
>>> clf = SGDClassifier(max_iter=10, tol=1e-3)
>>> clf.fit(X_features, y)
SGDClassifier(max_iter=10)
>>> clf.score(X_features, y)
1.0
"""
_parameter_constraints: dict = {
"skewedness": [Interval(Real, None, None, closed="neither")],
"n_components": [Interval(Integral, 1, None, closed="left")],
"random_state": ["random_state"],
}
def __init__(self, *, skewedness=1.0, n_components=100, random_state=None):
self.skewedness = skewedness
self.n_components = n_components
self.random_state = random_state
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit the model with X.
Samples random projection according to n_features.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : array-like, shape (n_samples,) or (n_samples, n_outputs), \
default=None
Target values (None for unsupervised transformations).
Returns
-------
self : object
Returns the instance itself.
"""
X = validate_data(self, X)
random_state = check_random_state(self.random_state)
n_features = X.shape[1]
uniform = random_state.uniform(size=(n_features, self.n_components))
# transform by inverse CDF of sech
self.random_weights_ = 1.0 / np.pi * np.log(np.tan(np.pi / 2.0 * uniform))
self.random_offset_ = random_state.uniform(0, 2 * np.pi, size=self.n_components)
if X.dtype == np.float32:
# Setting the data type of the fitted attribute will ensure the
# output data type during `transform`.
self.random_weights_ = self.random_weights_.astype(X.dtype, copy=False)
self.random_offset_ = self.random_offset_.astype(X.dtype, copy=False)
self._n_features_out = self.n_components
return self
def transform(self, X):
"""Apply the approximate feature map to X.
Parameters
----------
X : array-like, shape (n_samples, n_features)
New data, where `n_samples` is the number of samples
and `n_features` is the number of features. All values of X must be
strictly greater than "-skewedness".
Returns
-------
X_new : array-like, shape (n_samples, n_components)
Returns the instance itself.
"""
check_is_fitted(self)
X = validate_data(
self, X, copy=True, dtype=[np.float64, np.float32], reset=False
)
if (X <= -self.skewedness).any():
raise ValueError("X may not contain entries smaller than -skewedness.")
X += self.skewedness
np.log(X, X)
projection = safe_sparse_dot(X, self.random_weights_)
projection += self.random_offset_
np.cos(projection, projection)
projection *= np.sqrt(2.0) / np.sqrt(self.n_components)
return projection
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.transformer_tags.preserves_dtype = ["float64", "float32"]
return tags
class AdditiveChi2Sampler(TransformerMixin, BaseEstimator):
"""Approximate feature map for additive chi2 kernel.
Uses sampling the fourier transform of the kernel characteristic
at regular intervals.
Since the kernel that is to be approximated is additive, the components of
the input vectors can be treated separately. Each entry in the original
space is transformed into 2*sample_steps-1 features, where sample_steps is
a parameter of the method. Typical values of sample_steps include 1, 2 and
3.
Optimal choices for the sampling interval for certain data ranges can be
computed (see the reference). The default values should be reasonable.
Read more in the :ref:`User Guide <additive_chi_kernel_approx>`.
Parameters
----------
sample_steps : int, default=2
Gives the number of (complex) sampling points.
sample_interval : float, default=None
Sampling interval. Must be specified when sample_steps not in {1,2,3}.
Attributes
----------
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
--------
SkewedChi2Sampler : A Fourier-approximation to a non-additive variant of
the chi squared kernel.
sklearn.metrics.pairwise.chi2_kernel : The exact chi squared kernel.
sklearn.metrics.pairwise.additive_chi2_kernel : The exact additive chi
squared kernel.
Notes
-----
This estimator approximates a slightly different version of the additive
chi squared kernel then ``metric.additive_chi2`` computes.
This estimator is stateless and does not need to be fitted. However, we
recommend to call :meth:`fit_transform` instead of :meth:`transform`, as
parameter validation is only performed in :meth:`fit`.
References
----------
See `"Efficient additive kernels via explicit feature maps"
<http://www.robots.ox.ac.uk/~vedaldi/assets/pubs/vedaldi11efficient.pdf>`_
A. Vedaldi and A. Zisserman, Pattern Analysis and Machine Intelligence,
2011
Examples
--------
>>> from sklearn.datasets import load_digits
>>> from sklearn.linear_model import SGDClassifier
>>> from sklearn.kernel_approximation import AdditiveChi2Sampler
>>> X, y = load_digits(return_X_y=True)
>>> chi2sampler = AdditiveChi2Sampler(sample_steps=2)
>>> X_transformed = chi2sampler.fit_transform(X, y)
>>> clf = SGDClassifier(max_iter=5, random_state=0, tol=1e-3)
>>> clf.fit(X_transformed, y)
SGDClassifier(max_iter=5, random_state=0)
>>> clf.score(X_transformed, y)
0.9499...
"""
_parameter_constraints: dict = {
"sample_steps": [Interval(Integral, 1, None, closed="left")],
"sample_interval": [Interval(Real, 0, None, closed="left"), None],
}
def __init__(self, *, sample_steps=2, sample_interval=None):
self.sample_steps = sample_steps
self.sample_interval = sample_interval
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Only validates estimator's parameters.
This method allows to: (i) validate the estimator's parameters and
(ii) be consistent with the scikit-learn transformer API.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : array-like, shape (n_samples,) or (n_samples, n_outputs), \
default=None
Target values (None for unsupervised transformations).
Returns
-------
self : object
Returns the transformer.
"""
X = validate_data(self, X, accept_sparse="csr", ensure_non_negative=True)
if self.sample_interval is None and self.sample_steps not in (1, 2, 3):
raise ValueError(
"If sample_steps is not in [1, 2, 3],"
" you need to provide sample_interval"
)
return self
def transform(self, X):
"""Apply approximate feature map to X.
Parameters
----------
X : {array-like, sparse matrix}, shape (n_samples, n_features)
Training data, where `n_samples` is the number of samples
and `n_features` is the number of features.
Returns
-------
X_new : {ndarray, sparse matrix}, \
shape = (n_samples, n_features * (2*sample_steps - 1))
Whether the return value is an array or sparse matrix depends on
the type of the input X.
"""
X = validate_data(
self, X, accept_sparse="csr", reset=False, ensure_non_negative=True
)
sparse = sp.issparse(X)
if self.sample_interval is None:
# See figure 2 c) of "Efficient additive kernels via explicit feature maps" # noqa
# <http://www.robots.ox.ac.uk/~vedaldi/assets/pubs/vedaldi11efficient.pdf>
# A. Vedaldi and A. Zisserman, Pattern Analysis and Machine Intelligence, # noqa
# 2011
if self.sample_steps == 1:
sample_interval = 0.8
elif self.sample_steps == 2:
sample_interval = 0.5
elif self.sample_steps == 3:
sample_interval = 0.4
else:
raise ValueError(
"If sample_steps is not in [1, 2, 3],"
" you need to provide sample_interval"
)
else:
sample_interval = self.sample_interval
# zeroth component
# 1/cosh = sech
# cosh(0) = 1.0
transf = self._transform_sparse if sparse else self._transform_dense
return transf(X, self.sample_steps, sample_interval)
def get_feature_names_out(self, input_features=None):
"""Get output feature names for transformation.
Parameters
----------
input_features : array-like of str or None, default=None
Only used to validate feature names with the names seen in :meth:`fit`.
Returns
-------
feature_names_out : ndarray of str objects
Transformed feature names.
"""
# Note that passing attributes="n_features_in_" forces check_is_fitted
# to check if the attribute is present. Otherwise it will pass on this
# stateless estimator (requires_fit=False)
check_is_fitted(self, attributes="n_features_in_")
input_features = _check_feature_names_in(
self, input_features, generate_names=True
)
est_name = self.__class__.__name__.lower()
names_list = [f"{est_name}_{name}_sqrt" for name in input_features]
for j in range(1, self.sample_steps):
cos_names = [f"{est_name}_{name}_cos{j}" for name in input_features]
sin_names = [f"{est_name}_{name}_sin{j}" for name in input_features]
names_list.extend(cos_names + sin_names)
return np.asarray(names_list, dtype=object)
@staticmethod
def _transform_dense(X, sample_steps, sample_interval):
non_zero = X != 0.0
X_nz = X[non_zero]
X_step = np.zeros_like(X)
X_step[non_zero] = np.sqrt(X_nz * sample_interval)
X_new = [X_step]
log_step_nz = sample_interval * np.log(X_nz)
step_nz = 2 * X_nz * sample_interval
for j in range(1, sample_steps):
factor_nz = np.sqrt(step_nz / np.cosh(np.pi * j * sample_interval))
X_step = np.zeros_like(X)
X_step[non_zero] = factor_nz * np.cos(j * log_step_nz)
X_new.append(X_step)
X_step = np.zeros_like(X)
X_step[non_zero] = factor_nz * np.sin(j * log_step_nz)
X_new.append(X_step)
return np.hstack(X_new)
@staticmethod
def _transform_sparse(X, sample_steps, sample_interval):
indices = X.indices.copy()
indptr = X.indptr.copy()
data_step = np.sqrt(X.data * sample_interval)
X_step = sp.csr_matrix(
(data_step, indices, indptr), shape=X.shape, dtype=X.dtype, copy=False
)
X_new = [X_step]
log_step_nz = sample_interval * np.log(X.data)
step_nz = 2 * X.data * sample_interval
for j in range(1, sample_steps):
factor_nz = np.sqrt(step_nz / np.cosh(np.pi * j * sample_interval))
data_step = factor_nz * np.cos(j * log_step_nz)
X_step = sp.csr_matrix(
(data_step, indices, indptr), shape=X.shape, dtype=X.dtype, copy=False
)
X_new.append(X_step)
data_step = factor_nz * np.sin(j * log_step_nz)
X_step = sp.csr_matrix(
(data_step, indices, indptr), shape=X.shape, dtype=X.dtype, copy=False
)
X_new.append(X_step)
return sp.hstack(X_new)
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.requires_fit = False
tags.input_tags.positive_only = True
tags.input_tags.sparse = True
return tags
class Nystroem(ClassNamePrefixFeaturesOutMixin, TransformerMixin, BaseEstimator):
"""Approximate a kernel map using a subset of the training data.
Constructs an approximate feature map for an arbitrary kernel
using a subset of the data as basis.
Read more in the :ref:`User Guide <nystroem_kernel_approx>`.
.. versionadded:: 0.13
Parameters
----------
kernel : str or callable, default='rbf'
Kernel map to be approximated. A callable should accept two arguments
and the keyword arguments passed to this object as `kernel_params`, and
should return a floating point number.
gamma : float, default=None
Gamma parameter for the RBF, laplacian, polynomial, exponential chi2
and sigmoid kernels. Interpretation of the default value is left to
the kernel; see the documentation for sklearn.metrics.pairwise.
Ignored by other kernels.
coef0 : float, default=None
Zero coefficient for polynomial and sigmoid kernels.
Ignored by other kernels.
degree : float, default=None
Degree of the polynomial kernel. Ignored by other kernels.
kernel_params : dict, default=None
Additional parameters (keyword arguments) for kernel function passed
as callable object.
n_components : int, default=100
Number of features to construct.
How many data points will be used to construct the mapping.
random_state : int, RandomState instance or None, default=None
Pseudo-random number generator to control the uniform sampling without
replacement of `n_components` of the training data to construct the
basis kernel.
Pass an int for reproducible output across multiple function calls.
See :term:`Glossary <random_state>`.
n_jobs : int, default=None
The number of jobs to use for the computation. This works by breaking
down the kernel matrix into `n_jobs` even slices and computing them in
parallel.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
.. versionadded:: 0.24
Attributes
----------
components_ : ndarray of shape (n_components, n_features)
Subset of training points used to construct the feature map.
component_indices_ : ndarray of shape (n_components)
Indices of ``components_`` in the training set.
normalization_ : ndarray of shape (n_components, n_components)
Normalization matrix needed for embedding.
Square root of the kernel matrix on ``components_``.
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
--------
AdditiveChi2Sampler : Approximate feature map for additive chi2 kernel.
PolynomialCountSketch : Polynomial kernel approximation via Tensor Sketch.
RBFSampler : Approximate a RBF kernel feature map using random Fourier
features.
SkewedChi2Sampler : Approximate feature map for "skewed chi-squared" kernel.
sklearn.metrics.pairwise.kernel_metrics : List of built-in kernels.
References
----------
* Williams, C.K.I. and Seeger, M.
"Using the Nystroem method to speed up kernel machines",
Advances in neural information processing systems 2001
* T. Yang, Y. Li, M. Mahdavi, R. Jin and Z. Zhou
"Nystroem Method vs Random Fourier Features: A Theoretical and Empirical
Comparison",
Advances in Neural Information Processing Systems 2012
Examples
--------
>>> from sklearn import datasets, svm
>>> from sklearn.kernel_approximation import Nystroem
>>> X, y = datasets.load_digits(n_class=9, return_X_y=True)
>>> data = X / 16.
>>> clf = svm.LinearSVC()
>>> feature_map_nystroem = Nystroem(gamma=.2,
... random_state=1,
... n_components=300)
>>> data_transformed = feature_map_nystroem.fit_transform(data)
>>> clf.fit(data_transformed, y)
LinearSVC()
>>> clf.score(data_transformed, y)
0.9987...
"""
_parameter_constraints: dict = {
"kernel": [
StrOptions(set(PAIRWISE_KERNEL_FUNCTIONS.keys()) | {"precomputed"}),
callable,
],
"gamma": [Interval(Real, 0, None, closed="left"), None],
"coef0": [Interval(Real, None, None, closed="neither"), None],
"degree": [Interval(Real, 1, None, closed="left"), None],
"kernel_params": [dict, None],
"n_components": [Interval(Integral, 1, None, closed="left")],
"random_state": ["random_state"],
"n_jobs": [Integral, None],
}
def __init__(
self,
kernel="rbf",
*,
gamma=None,
coef0=None,
degree=None,
kernel_params=None,
n_components=100,
random_state=None,
n_jobs=None,
):
self.kernel = kernel
self.gamma = gamma
self.coef0 = coef0
self.degree = degree
self.kernel_params = kernel_params
self.n_components = n_components
self.random_state = random_state
self.n_jobs = n_jobs
@_fit_context(prefer_skip_nested_validation=True)
def fit(self, X, y=None):
"""Fit estimator to data.
Samples a subset of training points, computes kernel
on these and computes normalization matrix.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where `n_samples` is the number of samples
and `n_features` is the number of features.
y : array-like, shape (n_samples,) or (n_samples, n_outputs), \
default=None
Target values (None for unsupervised transformations).
Returns
-------
self : object
Returns the instance itself.
"""
X = validate_data(self, X, accept_sparse="csr")
rnd = check_random_state(self.random_state)
n_samples = X.shape[0]
# get basis vectors
if self.n_components > n_samples:
# XXX should we just bail?
n_components = n_samples
warnings.warn(
"n_components > n_samples. This is not possible.\n"
"n_components was set to n_samples, which results"
" in inefficient evaluation of the full kernel."
)
else:
n_components = self.n_components
n_components = min(n_samples, n_components)
inds = rnd.permutation(n_samples)
basis_inds = inds[:n_components]
basis = X[basis_inds]
basis_kernel = pairwise_kernels(
basis,
metric=self.kernel,
filter_params=True,
n_jobs=self.n_jobs,
**self._get_kernel_params(),
)
# sqrt of kernel matrix on basis vectors
U, S, V = svd(basis_kernel)
S = np.maximum(S, 1e-12)
self.normalization_ = np.dot(U / np.sqrt(S), V)
self.components_ = basis
self.component_indices_ = basis_inds
self._n_features_out = n_components
return self
def transform(self, X):
"""Apply feature map to X.
Computes an approximate feature map using the kernel
between some training points and X.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Data to transform.
Returns
-------
X_transformed : ndarray of shape (n_samples, n_components)
Transformed data.
"""
check_is_fitted(self)
X = validate_data(self, X, accept_sparse="csr", reset=False)
kernel_params = self._get_kernel_params()
embedded = pairwise_kernels(
X,
self.components_,
metric=self.kernel,
filter_params=True,
n_jobs=self.n_jobs,
**kernel_params,
)
return np.dot(embedded, self.normalization_.T)
def _get_kernel_params(self):
params = self.kernel_params
if params is None:
params = {}
if not callable(self.kernel) and self.kernel != "precomputed":
for param in KERNEL_PARAMS[self.kernel]:
if getattr(self, param) is not None:
params[param] = getattr(self, param)
else:
if (
self.gamma is not None
or self.coef0 is not None
or self.degree is not None
):
raise ValueError(
"Don't pass gamma, coef0 or degree to "
"Nystroem if using a callable "
"or precomputed kernel"
)
return params
def __sklearn_tags__(self):
tags = super().__sklearn_tags__()
tags.input_tags.sparse = True
tags.transformer_tags.preserves_dtype = ["float64", "float32"]
return tags
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