metric_learn.MMC

class metric_learn.MMC(max_iter=100, max_proj=10000, convergence_threshold=0.001, init='identity', diagonal=False, diagonal_c=1.0, verbose=False, preprocessor=None, random_state=None)[source]

Mahalanobis Metric for Clustering (MMC)

MMC minimizes the sum of squared distances between similar points, while enforcing the sum of distances between dissimilar ones to be greater than one. This leads to a convex and, thus, local-minima-free optimization problem that can be solved efficiently. However, the algorithm involves the computation of eigenvalues, which is the main speed-bottleneck. Since it has initially been designed for clustering applications, one of the implicit assumptions of MMC is that all classes form a compact set, i.e., follow a unimodal distribution, which restricts the possible use-cases of this method. However, it is one of the earliest and a still often cited technique.

Read more in the User Guide.

Parameters:
max_iterint, optional (default=100)

Maximum number of iterations of the optimization procedure.

max_projint, optional (default=10000)

Maximum number of projection steps.

convergence_thresholdfloat, optional (default=1e-3)

Convergence threshold for the optimization procedure.

initstring or numpy array, optional (default=’identity’)

Initialization of the Mahalanobis matrix. Possible options are ‘identity’, ‘covariance’, ‘random’, and a numpy array of shape (n_features, n_features).

‘identity’

An identity matrix of shape (n_features, n_features).

‘covariance’

The (pseudo-)inverse of the covariance matrix.

‘random’

The initial Mahalanobis matrix will be a random SPD matrix of shape (n_features, n_features), generated using sklearn.datasets.make_spd_matrix.

numpy array

An SPD matrix of shape (n_features, n_features), that will be used as such to initialize the metric.

diagonalbool, optional (default=False)

If True, a diagonal metric will be learned, i.e., a simple scaling of dimensions. The initialization will then be the diagonal coefficients of the matrix given as ‘init’.

diagonal_cfloat, optional (default=1.0)

Weight of the dissimilarity constraint for diagonal metric learning. Ignored if diagonal=False.

verbosebool, optional (default=False)

If True, prints information while learning

preprocessorarray-like, shape=(n_samples, n_features) or callable

The preprocessor to call to get tuples from indices. If array-like, tuples will be gotten like this: X[indices].

random_stateint or numpy.RandomState or None, optional (default=None)

A pseudo random number generator object or a seed for it if int. If init='random', random_state is used to initialize the random transformation.

See also

metric_learn.MMC
The original weakly-supervised algorithm
Supervised versions of weakly-supervised algorithms
The section of the project documentation that describes the supervised version of weakly supervised estimators.

References

[1]Xing, Jordan, Russell, Ng. Distance metric learning with application to clustering with side-information. NIPS 2002.

Examples

>>> from metric_learn import MMC
>>> pairs = [[[1.2, 7.5], [1.3, 1.5]],
>>>          [[6.4, 2.6], [6.2, 9.7]],
>>>          [[1.3, 4.5], [3.2, 4.6]],
>>>          [[6.2, 5.5], [5.4, 5.4]]]
>>> y = [1, 1, -1, -1]
>>> # in this task we want points where the first feature is close to be
>>> # closer to each other, no matter how close the second feature is
>>> mmc = MMC()
>>> mmc.fit(pairs, y)
Attributes:
n_iter_int

The number of iterations the solver has run.

components_numpy.ndarray, shape=(n_features, n_features)

The linear transformation L deduced from the learned Mahalanobis metric (See function components_from_metric.)

threshold_float

If the distance metric between two points is lower than this threshold, points will be classified as similar, otherwise they will be classified as dissimilar.

Methods

calibrate_threshold(pairs_valid, y_valid[, …]) Decision threshold calibration for pairwise binary classification
decision_function(pairs) Returns the decision function used to classify the pairs.
fit(pairs, y[, calibration_params]) Learn the MMC model.
get_mahalanobis_matrix() Returns a copy of the Mahalanobis matrix learned by the metric learner.
get_metric() Returns a function that takes as input two 1D arrays and outputs the learned metric score on these two points.
get_params([deep]) Get parameters for this estimator.
predict(pairs) Predicts the learned metric between input pairs.
score(pairs, y) Computes score of pairs similarity prediction.
score_pairs(pairs) Returns the learned Mahalanobis distance between pairs.
set_params(**params) Set the parameters of this estimator.
set_threshold(threshold) Sets the threshold of the metric learner to the given value threshold.
transform(X) Embeds data points in the learned linear embedding space.
__init__(max_iter=100, max_proj=10000, convergence_threshold=0.001, init='identity', diagonal=False, diagonal_c=1.0, verbose=False, preprocessor=None, random_state=None)

Initialize self. See help(type(self)) for accurate signature.

calibrate_threshold(pairs_valid, y_valid, strategy='accuracy', min_rate=None, beta=1.0)

Decision threshold calibration for pairwise binary classification

Method that calibrates the decision threshold (cutoff point) of the metric learner. This threshold will then be used when calling the method predict. The methods for picking cutoff points make use of traditional binary classification evaluation statistics such as the true positive and true negative rates and F-scores. The threshold will be found to maximize the chosen score on the validation set (pairs_valid, y_valid).

See more in the User Guide.

Parameters:
strategystr, optional (default=’accuracy’)

The strategy to use for choosing the cutoff threshold.

‘accuracy’

Selects a decision threshold that maximizes the accuracy.

‘f_beta’

Selects a decision threshold that maximizes the f_beta score, with beta given by the parameter beta.

‘max_tpr’

Selects a decision threshold that yields the highest true positive rate with true negative rate at least equal to the value of the parameter min_rate.

‘max_tnr’

Selects a decision threshold that yields the highest true negative rate with true positive rate at least equal to the value of the parameter min_rate.

betafloat in [0, 1], optional (default=None)

Beta value to be used in case strategy == ‘f_beta’.

min_ratefloat in [0, 1] or None, (default=None)

In case strategy is ‘max_tpr’ or ‘max_tnr’ this parameter must be set to specify the minimal value for the true negative rate or true positive rate respectively that needs to be achieved.

pairs_validarray-like, shape=(n_pairs_valid, 2, n_features)

The validation set of pairs to use to set the threshold.

y_validarray-like, shape=(n_pairs_valid,)

The labels of the pairs of the validation set to use to set the threshold. They must be +1 for positive pairs and -1 for negative pairs.

See also

sklearn.calibration
scikit-learn’s module for calibrating classifiers

References

[1]Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine, MH Zweig, G Campbell - Clinical chemistry, 1993
[2]most of the code of this function is from scikit-learn’s PR #10117
decision_function(pairs)

Returns the decision function used to classify the pairs.

Returns the opposite of the learned metric value between samples in every pair, to be consistent with scikit-learn conventions. Hence it should ideally be low for dissimilar samples and high for similar samples. This is the decision function that is used to classify pairs as similar (+1), or dissimilar (-1).

Parameters:
pairsarray-like, shape=(n_pairs, 2, n_features) or (n_pairs, 2)

3D Array of pairs to predict, with each row corresponding to two points, or 2D array of indices of pairs if the metric learner uses a preprocessor.

Returns:
y_predictednumpy.ndarray of floats, shape=(n_constraints,)

The predicted decision function value for each pair.

fit(pairs, y, calibration_params=None)[source]

Learn the MMC model.

The threshold will be calibrated on the trainset using the parameters calibration_params.

Parameters:
pairsarray-like, shape=(n_constraints, 2, n_features) or (n_constraints, 2)

3D Array of pairs with each row corresponding to two points, or 2D array of indices of pairs if the metric learner uses a preprocessor.

yarray-like, of shape (n_constraints,)

Labels of constraints. Should be -1 for dissimilar pair, 1 for similar.

calibration_paramsdict or None

Dictionary of parameters to give to calibrate_threshold for the threshold calibration step done at the end of fit. If None is given, calibrate_threshold will use the default parameters.

Returns:
selfobject

Returns the instance.

get_mahalanobis_matrix()

Returns a copy of the Mahalanobis matrix learned by the metric learner.

Returns:
Mnumpy.ndarray, shape=(n_features, n_features)

The copy of the learned Mahalanobis matrix.

get_metric()

Returns a function that takes as input two 1D arrays and outputs the learned metric score on these two points.

This function will be independent from the metric learner that learned it (it will not be modified if the initial metric learner is modified), and it can be directly plugged into the metric argument of scikit-learn’s estimators.

Returns:
metric_funfunction

The function described above.

See also

score_pairs
a method that returns the metric score between several pairs of points. Unlike get_metric, this is a method of the metric learner and therefore can change if the metric learner changes. Besides, it can use the metric learner’s preprocessor, and works on concatenated arrays.

Examples

>>> from metric_learn import NCA
>>> from sklearn.datasets import make_classification
>>> from sklearn.neighbors import KNeighborsClassifier
>>> nca = NCA()
>>> X, y = make_classification()
>>> nca.fit(X, y)
>>> knn = KNeighborsClassifier(metric=nca.get_metric())
>>> knn.fit(X, y) 
KNeighborsClassifier(algorithm='auto', leaf_size=30,
  metric=<function MahalanobisMixin.get_metric.<locals>.metric_fun
          at 0x...>,
  metric_params=None, n_jobs=None, n_neighbors=5, p=2,
  weights='uniform')
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:
paramsmapping of string to any

Parameter names mapped to their values.

predict(pairs)

Predicts the learned metric between input pairs. (For now it just calls decision function).

Returns the learned metric value between samples in every pair. It should ideally be low for similar samples and high for dissimilar samples.

Parameters:
pairsarray-like, shape=(n_pairs, 2, n_features) or (n_pairs, 2)

3D Array of pairs to predict, with each row corresponding to two points, or 2D array of indices of pairs if the metric learner uses a preprocessor.

Returns:
y_predictednumpy.ndarray of floats, shape=(n_constraints,)

The predicted learned metric value between samples in every pair.

score(pairs, y)

Computes score of pairs similarity prediction.

Returns the roc_auc score of the fitted metric learner. It is computed in the following way: for every value of a threshold t we classify all pairs of samples where the predicted distance is inferior to t as belonging to the “similar” class, and the other as belonging to the “dissimilar” class, and we count false positive and true positives as in a classical roc_auc curve.

Parameters:
pairsarray-like, shape=(n_pairs, 2, n_features) or (n_pairs, 2)

3D Array of pairs, with each row corresponding to two points, or 2D array of indices of pairs if the metric learner uses a preprocessor.

yarray-like, shape=(n_constraints,)

The corresponding labels.

Returns:
scorefloat

The roc_auc score.

score_pairs(pairs)

Returns the learned Mahalanobis distance between pairs.

This distance is defined as: \(d_M(x, x') = \sqrt{(x-x')^T M (x-x')}\) where M is the learned Mahalanobis matrix, for every pair of points x and x'. This corresponds to the euclidean distance between embeddings of the points in a new space, obtained through a linear transformation. Indeed, we have also: \(d_M(x, x') = \sqrt{(x_e - x_e')^T (x_e- x_e')}\), with \(x_e = L x\) (See MahalanobisMixin).

Parameters:
pairsarray-like, shape=(n_pairs, 2, n_features) or (n_pairs, 2)

3D Array of pairs to score, with each row corresponding to two points, for 2D array of indices of pairs if the metric learner uses a preprocessor.

Returns:
scoresnumpy.ndarray of shape=(n_pairs,)

The learned Mahalanobis distance for every pair.

See also

get_metric
a method that returns a function to compute the metric between two points. The difference with score_pairs is that it works on two 1D arrays and cannot use a preprocessor. Besides, the returned function is independent of the metric learner and hence is not modified if the metric learner is.
Mahalanobis Distances
The section of the project documentation that describes Mahalanobis Distances.
set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). 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:
selfobject

Estimator instance.

set_threshold(threshold)

Sets the threshold of the metric learner to the given value threshold.

See more in the User Guide.

Parameters:
thresholdfloat

The threshold value we want to set. It is the value to which the predicted distance for test pairs will be compared. If they are superior to the threshold they will be classified as similar (+1), and dissimilar (-1) if not.

Returns:
self_PairsClassifier

The pairs classifier with the new threshold set.

transform(X)

Embeds data points in the learned linear embedding space.

Transforms samples in X into X_embedded, samples inside a new embedding space such that: X_embedded = X.dot(L.T), where L is the learned linear transformation (See MahalanobisMixin).

Parameters:
Xnumpy.ndarray, shape=(n_samples, n_features)

The data points to embed.

Returns:
X_embeddednumpy.ndarray, shape=(n_samples, n_components)

The embedded data points.