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qml.kernels.target_alignment

target_alignment(X, Y, kernel, assume_normalized_kernel=False, rescale_class_labels=True)[source]

Target alignment of a given kernel function.

This function is an alias for polarity() with normalize=True.

For a dataset with feature vectors \(\{x_i\}\) and associated labels \(\{y_i\}\), the target alignment of the kernel function \(k\) is given by

\[\operatorname{TA}(k) = \frac{\sum_{i,j=1}^n y_i y_j k(x_i, x_j)} {\sqrt{\sum_{i,j=1}^n y_i y_j} \sqrt{\sum_{i,j=1}^n k(x_i, x_j)^2}}\]

If the dataset is unbalanced, that is if the numbers of datapoints in the two classes \(n_+\) and \(n_-\) differ, rescale_class_labels=True will apply a rescaling according to \(\tilde{y}_i = \frac{y_i}{n_{y_i}}\). This is activated by default and only results in a prefactor that depends on the size of the dataset for balanced datasets.

Parameters:

  • 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:

The kernel-target alignment.

Return type:

float

Example:

Consider a simple kernel function based on AngleEmbedding:

dev = qml.device('default.qubit', wires=2)
@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

>>> rng = np.random.default_rng(seed=1234)
>>> X = rng.random((4, 2))
>>> Y = np.array([-1, -1, 1, 1])
>>> qml.kernels.target_alignment(X, Y, kernel)
np.float64(0.0582...)

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)
>>> print(np.isclose(target_alignment, normalized_polarity))
True
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