"""
The :mod:`sklearn.kernel_approximation` module implements several
approximate kernel feature maps based on Fourier transforms and Count Sketches.
"""
# Author: Andreas Mueller <amueller@ais.uni-bonn.de>
# Daniel Lopez-Sanchez (TensorSketch) <lope@usal.es>
# License: BSD 3 clause
import warnings
import numpy as np
import scipy.sparse as sp
from scipy.linalg import svd
try:
from scipy.fft import fft, ifft
except ImportError: # scipy < 1.4
from scipy.fftpack import fft, ifft
from .base import BaseEstimator
from .base import TransformerMixin
from .utils import check_random_state, as_float_array
from .utils.extmath import safe_sparse_dot
from .utils.validation import check_is_fitted
from .metrics.pairwise import pairwise_kernels, KERNEL_PARAMS
from .utils.validation import check_non_negative, _deprecate_positional_args
class PolynomialCountSketch(BaseEstimator, TransformerMixin):
"""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.
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
"""
def __init__(self, *, gamma=1., 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
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 in the number of samples
and n_features is the number of features.
Returns
-------
self : object
Returns the transformer.
"""
if not self.degree >= 1:
raise ValueError(f"degree={self.degree} should be >=1.")
X = self._validate_data(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))
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 in the number of samples
and n_features is the number of features.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
"""
check_is_fitted(self)
X = self._validate_data(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
class RBFSampler(TransformerMixin, BaseEstimator):
"""Approximates feature map of an RBF kernel by Monte Carlo approximation
of its Fourier transform.
It implements a variant of Random Kitchen Sinks.[1]
Read more in the :ref:`User Guide <rbf_kernel_approx>`.
Parameters
----------
gamma : float, default=1.0
Parameter of RBF kernel: exp(-gamma * x^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=float64
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=float64
Random projection directions drawn from the Fourier transform
of the RBF kernel.
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
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)
"""
@_deprecate_positional_args
def __init__(self, *, gamma=1., n_components=100, random_state=None):
self.gamma = gamma
self.n_components = n_components
self.random_state = random_state
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 in the number of samples
and n_features is the number of features.
Returns
-------
self : object
Returns the transformer.
"""
X = self._validate_data(X, accept_sparse='csr')
random_state = check_random_state(self.random_state)
n_features = X.shape[1]
self.random_weights_ = (np.sqrt(2 * self.gamma) * random_state.normal(
size=(n_features, self.n_components)))
self.random_offset_ = random_state.uniform(0, 2 * np.pi,
size=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 in the number of samples
and n_features is the number of features.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
"""
check_is_fitted(self)
X = self._validate_data(X, accept_sparse='csr', reset=False)
projection = safe_sparse_dot(X, self.random_weights_)
projection += self.random_offset_
np.cos(projection, projection)
projection *= np.sqrt(2.) / np.sqrt(self.n_components)
return projection
class SkewedChi2Sampler(TransformerMixin, BaseEstimator):
"""Approximates feature map of the "skewed chi-squared" kernel by Monte
Carlo approximation of its Fourier transform.
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.
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
References
----------
See "Random Fourier Approximations for Skewed Multiplicative Histogram
Kernels" by Fuxin Li, Catalin Ionescu and Cristian Sminchisescu.
See Also
--------
AdditiveChi2Sampler : A different approach for approximating an additive
variant of the chi squared kernel.
sklearn.metrics.pairwise.chi2_kernel : The exact chi squared kernel.
"""
@_deprecate_positional_args
def __init__(self, *, skewedness=1., n_components=100, random_state=None):
self.skewedness = skewedness
self.n_components = n_components
self.random_state = random_state
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 in the number of samples
and n_features is the number of features.
Returns
-------
self : object
Returns the transformer.
"""
X = self._validate_data(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. / np.pi
* np.log(np.tan(np.pi / 2. * uniform)))
self.random_offset_ = random_state.uniform(0, 2 * np.pi,
size=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 in 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)
"""
check_is_fitted(self)
X = as_float_array(X, copy=True)
X = self._validate_data(X, copy=False, 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.) / np.sqrt(self.n_components)
return projection
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
----------
sample_interval_ : float
Stored sampling interval. Specified as a parameter if sample_steps not
in {1,2,3}.
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...
Notes
-----
This estimator approximates a slightly different version of the additive
chi squared kernel then ``metric.additive_chi2`` computes.
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.
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
"""
@_deprecate_positional_args
def __init__(self, *, sample_steps=2, sample_interval=None):
self.sample_steps = sample_steps
self.sample_interval = sample_interval
def fit(self, X, y=None):
"""Set the parameters
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples in the number of samples
and n_features is the number of features.
Returns
-------
self : object
Returns the transformer.
"""
X = self._validate_data(X, accept_sparse='csr')
check_non_negative(X, 'X in AdditiveChi2Sampler.fit')
if self.sample_interval is None:
# See reference, figure 2 c)
if self.sample_steps == 1:
self.sample_interval_ = 0.8
elif self.sample_steps == 2:
self.sample_interval_ = 0.5
elif self.sample_steps == 3:
self.sample_interval_ = 0.4
else:
raise ValueError("If sample_steps is not in [1, 2, 3],"
" you need to provide sample_interval")
else:
self.sample_interval_ = self.sample_interval
return self
def transform(self, X):
"""Apply approximate feature map to X.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
Returns
-------
X_new : {ndarray, sparse matrix}, \
shape = (n_samples, n_features * (2*sample_steps + 1))
Whether the return value is an array of sparse matrix depends on
the type of the input X.
"""
msg = ("%(name)s is not fitted. Call fit to set the parameters before"
" calling transform")
check_is_fitted(self, msg=msg)
X = self._validate_data(X, accept_sparse='csr', reset=False)
check_non_negative(X, 'X in AdditiveChi2Sampler.transform')
sparse = sp.issparse(X)
# zeroth component
# 1/cosh = sech
# cosh(0) = 1.0
transf = self._transform_sparse if sparse else self._transform_dense
return transf(X)
def _transform_dense(self, X):
non_zero = (X != 0.0)
X_nz = X[non_zero]
X_step = np.zeros_like(X)
X_step[non_zero] = np.sqrt(X_nz * self.sample_interval_)
X_new = [X_step]
log_step_nz = self.sample_interval_ * np.log(X_nz)
step_nz = 2 * X_nz * self.sample_interval_
for j in range(1, self.sample_steps):
factor_nz = np.sqrt(step_nz /
np.cosh(np.pi * j * self.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)
def _transform_sparse(self, X):
indices = X.indices.copy()
indptr = X.indptr.copy()
data_step = np.sqrt(X.data * self.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 = self.sample_interval_ * np.log(X.data)
step_nz = 2 * X.data * self.sample_interval_
for j in range(1, self.sample_steps):
factor_nz = np.sqrt(step_nz /
np.cosh(np.pi * j * self.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 _more_tags(self):
return {'stateless': True,
'requires_positive_X': True}
class Nystroem(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 : string 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_``.
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...
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
See Also
--------
RBFSampler : An approximation to the RBF kernel using random Fourier
features.
sklearn.metrics.pairwise.kernel_metrics : List of built-in kernels.
"""
@_deprecate_positional_args
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
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 of shape (n_samples, n_features)
Training data.
"""
X = self._validate_data(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_ = inds
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 = self._validate_data(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 _more_tags(self):
return {
'_xfail_checks': {
'check_transformer_preserve_dtypes':
'dtypes are preserved but not at a close enough precision',
},
'preserves_dtype': [np.float64, np.float32]
}