MMD_RKHS

class mlquantify.mixture.MMD_RKHS(kernel='rbf', gamma=None, degree=3, coef0=0.0)

Maximum Mean Discrepancy in RKHS (MMD-RKHS) quantification method.

This method estimates class prevalences in an unlabeled test set by matching the kernel mean embedding of the test distribution to a convex combination of the class-conditional training embeddings.

Let \(\mathcal{X} \subseteq \mathbb{R}^d\) be the input space and \(\mathcal{Y} = \{0, \dots, C-1\}\) the label set. Let \(K\) be a positive definite kernel with RKHS \(\mathcal{H}\) and feature map \(\phi\), so that \(K(x, x') = \langle \phi(x), \phi(x') \rangle_{\mathcal{H}}\).

For each class \(y\), the class-conditional kernel mean embedding is

\[\mu_y \;=\; \mathbb{E}_{x \sim P_{D}(x \mid y)}[\phi(x)] \in \mathcal{H},\]

and the test mean embedding is

\[\mu_U \;=\; \mathbb{E}_{x \sim P_{U}(x)}[\phi(x)] \in \mathcal{H}.\]

Under prior probability shift, the test distribution satisfies

\[P_U(x) = \sum_{y=0}^{C-1} \theta_y \, P_D(x \mid y),\]

which implies

\[\mu_U = \sum_{y=0}^{C-1} \theta_y \, \mu_y,\]

where \(\theta \in \Delta^{C-1}\) is the class prevalence vector. The MMD-RKHS estimator solves

\[\hat{\theta} \;=\; \arg\min_{\theta \in \Delta^{C-1}} \big\lVert \textstyle\sum_{y=0}^{C-1} \theta_y \mu_y - \mu_U \big\rVert_{\mathcal{H}}^2.\]

In practice, embeddings are approximated by empirical means. Using the kernel trick, the objective can be written as a quadratic program

\[\hat{\theta} \;=\; \arg\min_{\theta \in \Delta^{C-1}} \big( \theta^\top G \, \theta - 2 \, h^\top \theta \big),\]

with

\[G_{yy'} = \langle \hat{\mu}_y, \hat{\mu}_{y'} \rangle_{\mathcal{H}}, \qquad h_y = \langle \hat{\mu}_y, \hat{\mu}_U \rangle_{\mathcal{H}}.\]

The solution \(\hat{\theta}\) is the estimated prevalence vector.

Parameters:

kernel{‘rbf’, ‘linear’, ‘poly’, ‘sigmoid’, ‘cosine’}, default=’rbf’

Kernel used to build the RKHS where MMD is computed.

gammafloat or None, default=None

Kernel coefficient for ‘rbf’ and ‘sigmoid’.

degreeint, default=3

Degree of the polynomial kernel.

coef0float, default=0.0

Independent term in ‘poly’ and ‘sigmoid’ kernels.

strategy{‘ovr’, ‘ovo’}, default=’ovr’

Multiclass quantification strategy flag (for consistency with other mixture-based quantifiers).

Attributes:

classes_ndarray of shape (n_classes,)

Class labels seen during fitting.

X_train_ndarray of shape (n_train, n_features)

Training feature matrix.

y_train_ndarray of shape (n_train,)

Training labels.

class_means_ndarray of shape (n_classes, n_train)

Empirical class-wise kernel mean embeddings in the span of training samples.

K_train_ndarray of shape (n_train, n_train)

Gram matrix of training samples under the chosen kernel.

References

[1]

Iyer, A., Nath, S., & Sarawagi, S. (2014). Maximum Mean Discrepancy for Class Ratio Estimation: Convergence Bounds and Kernel Selection. ICML.

[2]

Esuli, A., Moreo, A., & Sebastiani, F. (2023). Learning to Quantify. Springer.

best_mixture(X_test, X_train, y_train)

Implements the MMD-based class ratio estimation:

\[\min_{\theta \in \Delta^{C-1}} \| \sum_{y=0}^{C-1} \theta_y \mu_y - \mu_U \|^2\]

and returns (theta, objective_value).

fit(X, y, *args, **kwargs)

Fit the quantifier using the provided data and learner.

get_best_distance(*args, **kwargs)

Get the best distance value from the mixture fitting process.

Notes

If the quantifier has not been fitted yet, it will fit the model for getting the best distance.

classmethod get_distance(dist_train, dist_test, measure='hellinger')

Compute distance between two distributions.

get_metadata_routing()

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict

Parameter names mapped to their values.

predict(X, *args, **kwargs)

Predict class prevalences for the given data.

save_quantifier(path: str | None = None) → None

Save the quantifier instance to a file.

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict

Estimator parameters.

Returns:

selfestimator instance

Estimator instance.