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()
withnormalize=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 labelsY
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
inpolarity
:>>> 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)