# metric_learn.NCA¶

class metric_learn.NCA(init='auto', n_components=None, max_iter=100, tol=None, verbose=False, preprocessor=None, random_state=None)[source]

Neighborhood Components Analysis (NCA)

NCA is a distance metric learning algorithm which aims to improve the accuracy of nearest neighbors classification compared to the standard Euclidean distance. The algorithm directly maximizes a stochastic variant of the leave-one-out k-nearest neighbors(KNN) score on the training set. It can also learn a low-dimensional linear transformation of data that can be used for data visualization and fast classification.

Read more in the User Guide.

Parameters: initstring or numpy array, optional (default=’auto’)Initialization of the linear transformation. Possible options are ‘auto’, ‘pca’, ‘identity’, ‘random’, and a numpy array of shape (n_features_a, n_features_b). ‘auto’Depending on n_components, the most reasonable initialization will be chosen. If n_components <= n_classes we use ‘lda’, as it uses labels information. If not, but n_components < min(n_features, n_samples), we use ‘pca’, as it projects data in meaningful directions (those of higher variance). Otherwise, we just use ‘identity’. ‘pca’n_components principal components of the inputs passed to fit() will be used to initialize the transformation. (See sklearn.decomposition.PCA) ‘lda’min(n_components, n_classes) most discriminative components of the inputs passed to fit() will be used to initialize the transformation. (If n_components > n_classes, the rest of the components will be zero.) (See sklearn.discriminant_analysis.LinearDiscriminantAnalysis) ‘identity’If n_components is strictly smaller than the dimensionality of the inputs passed to fit(), the identity matrix will be truncated to the first n_components rows. ‘random’The initial transformation will be a random array of shape (n_components, n_features). Each value is sampled from the standard normal distribution. numpy arrayn_features_b must match the dimensionality of the inputs passed to fit() and n_features_a must be less than or equal to that. If n_components is not None, n_features_a must match it. n_componentsint or None, optional (default=None)Dimensionality of reduced space (if None, defaults to dimension of X). max_iterint, optional (default=100)Maximum number of iterations done by the optimization algorithm. tolfloat, optional (default=None)Convergence tolerance for the optimization. verbosebool, optional (default=False)Whether to print progress messages or not. 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. If init='pca', random_state is passed as an argument to PCA when initializing the transformation.

References

 [1] J. Goldberger, G. Hinton, S. Roweis, R. Salakhutdinov. Neighbourhood Components Analysis. NIPS 2005.
 [2] Wikipedia entry on Neighborhood Components Analysis

Examples

>>> import numpy as np
>>> from metric_learn import NCA
>>> X = iris_data['data']
>>> Y = iris_data['target']
>>> nca = NCA(max_iter=1000)
>>> nca.fit(X, Y)

Attributes: n_iter_intThe number of iterations the solver has run. components_numpy.ndarray, shape=(n_components, n_features)The learned linear transformation L.

Methods

 fit(X, y) X: data matrix, (n x d) y: scalar labels, (n) fit_transform(X[, y]) Fit to data, then transform it. 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. score_pairs(pairs) Returns the learned Mahalanobis distance between pairs. set_params(**params) Set the parameters of this estimator. transform(X) Embeds data points in the learned linear embedding space.
__init__(init='auto', n_components=None, max_iter=100, tol=None, verbose=False, preprocessor=None, random_state=None)[source]

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

fit(X, y)[source]

X: data matrix, (n x d) y: scalar labels, (n)

fit_transform(X, y=None, **fit_params)

Fit to data, then transform it.

Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.

Parameters: X{array-like, sparse matrix, dataframe} of shape (n_samples, n_features) yndarray of shape (n_samples,), default=NoneTarget values. **fit_paramsdictAdditional fit parameters. X_newndarray array of shape (n_samples, n_features_new)Transformed array.
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_funfunctionThe function described above.

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=TrueIf True, will return the parameters for this estimator and contained subobjects that are estimators. paramsmapping of string to anyParameter names mapped to their values.
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. scoresnumpy.ndarray of shape=(n_pairs,)The learned Mahalanobis distance for every pair.

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: **paramsdictEstimator parameters. selfobjectEstimator instance.
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. X_embeddednumpy.ndarray, shape=(n_samples, n_components)The embedded data points.