Source code for pennylane.kernels.cost_functions
# Copyright 2018-2021 Xanadu Quantum Technologies Inc.
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# you may not use this file except in compliance with the License.
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# http://www.apache.org/licenses/LICENSE-2.0
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"""
This file contains functionalities for kernel related costs.
See `here <https://www.doi.org/10.1007/s10462-012-9369-4>`_ for a review.
"""
import numpy as np
from ..math import frobenius_inner_product
from .utils import square_kernel_matrix
[docs]def polarity(
X,
Y,
kernel,
assume_normalized_kernel=False,
rescale_class_labels=True,
normalize=False,
):
r"""Polarity of a given kernel function.
For a dataset with feature vectors :math:`\{x_i\}` and associated labels :math:`\{y_i\}`,
the polarity of the kernel function :math:`k` is given by
.. math ::
\operatorname{P}(k) = \sum_{i,j=1}^n y_i y_j k(x_i, x_j)
If the dataset is unbalanced, that is if the numbers of datapoints in the
two classes :math:`n_+` and :math:`n_-` differ,
``rescale_class_labels=True`` will apply a rescaling according to
:math:`\tilde{y}_i = \frac{y_i}{n_{y_i}}`. This is activated by default
and only results in a prefactor that depends on the size of the dataset
for balanced datasets.
The keyword argument ``assume_normalized_kernel`` is passed to
:func:`~.kernels.square_kernel_matrix`, for the computation
:func:`~.utils.frobenius_inner_product` is used.
Args:
X (list[datapoint]): List of datapoints.
Y (list[float]): List of class labels of datapoints, assumed to be either -1 or 1.
kernel ((datapoint, datapoint) -> float): Kernel function that maps datapoints to kernel value.
assume_normalized_kernel (bool, optional): Assume that the kernel is normalized, i.e.
the kernel evaluates to 1 when both arguments are the same datapoint.
rescale_class_labels (bool, optional): Rescale the class labels. This is important to take
care of unbalanced datasets.
normalize (bool): If True, rescale the polarity to the target_alignment.
Returns:
float: The kernel polarity.
**Example:**
Consider a simple kernel function based on :class:`~.templates.embeddings.AngleEmbedding`:
.. code-block :: python
dev = qml.device('default.qubit', wires=2, shots=None)
@qml.qnode(dev)
def circuit(x1, x2):
qml.templates.AngleEmbedding(x1, wires=dev.wires)
qml.adjoint(qml.templates.AngleEmbedding)(x2, wires=dev.wires)
return qml.probs(wires=dev.wires)
kernel = lambda x1, x2: circuit(x1, x2)[0]
We can then compute the polarity on a set of 4 (random) feature
vectors ``X`` with labels ``Y`` via
>>> X = np.random.random((4, 2))
>>> Y = np.array([-1, -1, 1, 1])
>>> qml.kernels.polarity(X, Y, kernel)
tensor(0.04361349, requires_grad=True)
"""
# pylint: disable=too-many-arguments
K = square_kernel_matrix(X, kernel, assume_normalized_kernel=assume_normalized_kernel)
if rescale_class_labels:
nplus = np.count_nonzero(np.array(Y) == 1)
nminus = len(Y) - nplus
_Y = np.array([y / nplus if y == 1 else y / nminus for y in Y])
else:
_Y = np.array(Y)
T = np.outer(_Y, _Y)
return frobenius_inner_product(K, T, normalize=normalize)
[docs]def target_alignment(
X,
Y,
kernel,
assume_normalized_kernel=False,
rescale_class_labels=True,
):
r"""Target alignment of a given kernel function.
This function is an alias for :func:`~.kernels.polarity` with ``normalize=True``.
For a dataset with feature vectors :math:`\{x_i\}` and associated labels :math:`\{y_i\}`, the
target alignment of the kernel function :math:`k` is given by
.. math ::
\operatorname{TA}(k) = \frac{\sum_{i,j=1}^n y_i y_j k(x_i, x_j)}
{\sqrt{\sum_{i,j=1}^n y_i y_j} \sqrt{\sum_{i,j=1}^n k(x_i, x_j)^2}}
If the dataset is unbalanced, that is if the numbers of datapoints in the
two classes :math:`n_+` and :math:`n_-` differ,
``rescale_class_labels=True`` will apply a rescaling according to
:math:`\tilde{y}_i = \frac{y_i}{n_{y_i}}`. This is activated by default
and only results in a prefactor that depends on the size of the dataset
for balanced datasets.
Args:
X (list[datapoint]): List of datapoints
Y (list[float]): List of class labels of datapoints, assumed to be either -1 or 1.
kernel ((datapoint, datapoint) -> float): Kernel function that maps datapoints to kernel value.
assume_normalized_kernel (bool, optional): Assume that the kernel is normalized, i.e.
the kernel evaluates to 1 when both arguments are the same datapoint.
rescale_class_labels (bool, optional): Rescale the class labels. This is important to take
care of unbalanced datasets.
Returns:
float: The kernel-target alignment.
**Example:**
Consider a simple kernel function based on :class:`~.templates.embeddings.AngleEmbedding`:
.. code-block :: python
dev = qml.device('default.qubit', wires=2, shots=None)
@qml.qnode(dev)
def circuit(x1, x2):
qml.templates.AngleEmbedding(x1, wires=dev.wires)
qml.adjoint(qml.templates.AngleEmbedding)(x2, wires=dev.wires)
return qml.probs(wires=dev.wires)
kernel = lambda x1, x2: circuit(x1, x2)[0]
We can then compute the kernel-target alignment on a set of 4 (random)
feature vectors ``X`` with labels ``Y`` via
>>> X = np.random.random((4, 2))
>>> Y = np.array([-1, -1, 1, 1])
>>> qml.kernels.target_alignment(X, Y, kernel)
tensor(0.01124802, requires_grad=True)
We can see that this is equivalent to using ``normalize=True`` in
``polarity``:
>>> target_alignment = qml.kernels.target_alignment(X, Y, kernel)
>>> normalized_polarity = qml.kernels.polarity(X, Y, kernel, normalize=True)
>>> np.isclose(target_alignment, normalized_polarity)
tensor(True, requires_grad=True)
"""
return polarity(
X,
Y,
kernel,
assume_normalized_kernel=assume_normalized_kernel,
rescale_class_labels=rescale_class_labels,
normalize=True,
)
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