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%matplotlib inline

Neighborhood Components Analysis Illustration

This example illustrates a learned distance metric that maximizes the nearest neighbors classification accuracy. It provides a visual representation of this metric compared to the original point space. Please refer to the User Guide <nca> for more information.

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# License: BSD 3 clause

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_classification
from sklearn.neighbors import NeighborhoodComponentsAnalysis
from matplotlib import cm
from scipy.special import logsumexp


Original points

First we create a data set of 9 samples from 3 classes, and plot the points in the original space. For this example, we focus on the classification of point no. 3. The thickness of a link between point no. 3 and another point is proportional to their distance.

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X, y = make_classification(n_samples=9, n_features=2, n_informative=2,
                           n_redundant=0, n_classes=3, n_clusters_per_class=1,
                           class_sep=1.0, random_state=0)

ax = plt.gca()
for i in range(X.shape[0]):
    ax.text(X[i, 0], X[i, 1], str(i), va='center', ha='center')
    ax.scatter(X[i, 0], X[i, 1], s=300, c=cm.Set1(y[[i]]), alpha=0.4)

ax.set_title("Original points")
ax.axis('equal')  # so that boundaries are displayed correctly as circles

def link_thickness_i(X, i):
    diff_embedded = X[i] - X
    dist_embedded = np.einsum('ij,ij->i', diff_embedded,
    dist_embedded[i] = np.inf

    # compute exponentiated distances (use the log-sum-exp trick to
    # avoid numerical instabilities
    exp_dist_embedded = np.exp(-dist_embedded -
    return exp_dist_embedded

def relate_point(X, i, ax):
    pt_i = X[i]
    for j, pt_j in enumerate(X):
        thickness = link_thickness_i(X, i)
        if i != j:
            line = ([pt_i[0], pt_j[0]], [pt_i[1], pt_j[1]])
            ax.plot(*line, c=cm.Set1(y[j]),

i = 3
relate_point(X, i, ax)

Learning an embedding

We use :class:~sklearn.neighbors.NeighborhoodComponentsAnalysis to learn an embedding and plot the points after the transformation. We then take the embedding and find the nearest neighbors.

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nca = NeighborhoodComponentsAnalysis(max_iter=30, random_state=0)
nca =, y)

ax2 = plt.gca()
X_embedded = nca.transform(X)
relate_point(X_embedded, i, ax2)

for i in range(len(X)):
    ax2.text(X_embedded[i, 0], X_embedded[i, 1], str(i),
             va='center', ha='center')
    ax2.scatter(X_embedded[i, 0], X_embedded[i, 1], s=300, c=cm.Set1(y[[i]]),

ax2.set_title("NCA embedding")