Source code for pennylane.kernels.utils

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

#     http://www.apache.org/licenses/LICENSE-2.0

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
This file contains functionalities that simplify working with kernels.
"""
from itertools import product
import pennylane as qml


[docs]def square_kernel_matrix(X, kernel, assume_normalized_kernel=False): r"""Computes the square matrix of pairwise kernel values for a given dataset. Args: X (list[datapoint]): List of datapoints kernel ((datapoint, datapoint) -> float): Kernel function that maps datapoints to kernel value. assume_normalized_kernel (bool, optional): Assume that the kernel is normalized, in which case the diagonal of the kernel matrix is set to 1, avoiding unnecessary computations. Returns: array[float]: The square matrix of kernel values. **Example:** Consider a simple kernel function based on :class:`~.templates.embeddings.AngleEmbedding`: .. code-block :: python dev = qml.device('default.qubit') @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 matrix on a set of 4 (random) feature vectors ``X`` via >>> X = np.random.random((4, 2)) >>> qml.kernels.square_kernel_matrix(X, kernel) tensor([[1. , 0.9532702 , 0.96864001, 0.90932897], [0.9532702 , 1. , 0.99727485, 0.95685561], [0.96864001, 0.99727485, 1. , 0.96605621], [0.90932897, 0.95685561, 0.96605621, 1. ]], requires_grad=True) """ N = qml.math.shape(X)[0] if assume_normalized_kernel and N == 1: return qml.math.eye(1, like=qml.math.get_interface(X)) matrix = [None] * N**2 # Compute all off-diagonal kernel values, using symmetry of the kernel matrix for i in range(N): for j in range(i + 1, N): matrix[N * i + j] = (kernel_value := kernel(X[i], X[j])) matrix[N * j + i] = kernel_value if assume_normalized_kernel: # Create a one-like entry that has the same interface and batching as the kernel output # As we excluded the case N=1 together with assume_normalized_kernel above, matrix[1] exists one = qml.math.ones_like(matrix[1]) for i in range(N): matrix[N * i + i] = one else: # Fill the diagonal by computing the corresponding kernel values for i in range(N): matrix[N * i + i] = kernel(X[i], X[i]) shape = (N, N) if qml.math.ndim(matrix[0]) == 0 else (N, N, qml.math.size(matrix[0])) return qml.math.moveaxis(qml.math.reshape(qml.math.stack(matrix), shape), -1, 0)
[docs]def kernel_matrix(X1, X2, kernel): r"""Computes the matrix of pairwise kernel values for two given datasets. Args: X1 (list[datapoint]): List of datapoints (first argument) X2 (list[datapoint]): List of datapoints (second argument) kernel ((datapoint, datapoint) -> float): Kernel function that maps datapoints to kernel value. Returns: array[float]: The matrix of kernel values. **Example:** Consider a simple kernel function based on :class:`~.templates.embeddings.AngleEmbedding`: .. code-block :: python dev = qml.device('default.qubit') @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] With this method we can systematically evaluate the kernel function ``kernel`` on pairs of datapoints, where the points stem from different datasets, like a training and a test dataset. >>> X_train = np.random.random((4,2)) >>> X_test = np.random.random((3,2)) >>> qml.kernels.kernel_matrix(X_train, X_test, kernel) tensor([[0.88875298, 0.90655175, 0.89926447], [0.93762197, 0.98163781, 0.93076383], [0.91977339, 0.9799841 , 0.91582698], [0.80376818, 0.98720925, 0.79349212]], requires_grad=True) As we can see, for :math:`n` and :math:`m` datapoints in the first and second dataset respectively, the output matrix has the shape :math:`n\times m`. """ N = qml.math.shape(X1)[0] M = qml.math.shape(X2)[0] matrix = qml.math.stack([kernel(x, y) for x, y in product(X1, X2)]) if qml.math.ndim(matrix[0]) == 0: return qml.math.reshape(matrix, (N, M)) return qml.math.moveaxis(qml.math.reshape(matrix, (N, M, qml.math.size(matrix[0]))), -1, 0)