Neighborhood Components Analysis — 9:31 min

• 9:31 min | Ultima modificación: Septiembre 24, 2021 | YouTube

Este transformador realiza una transformación óptima del dataset de entrenamiento, tal que maximiza la probabilidad p_i de que una muestra sea corectamente clasificada:

\arg \max_L \sum_i p_i

donde

p_i = \sum_{j \in C_i} p_{ij}

C_i es el conjunto de puntos que pertenencen a la misma clase del punto i, y p_{ij} es la función softmax:

p_{ij} = \frac{\exp(-||\text{L}(x_i - x_j)||^2} {\sum_{k \ne i} \exp(-||\text{L}(x_i - x_k)||^2 )}

https://scikit-learn.org/stable/auto_examples/neighbors/plot_nca_illustration.html

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

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,
)


plt.figure(figsize=(14, 7))

plt.subplot(1, 2, 1)

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

plt.gca().set_title("Original points")
plt.gca().axes.get_xaxis().set_visible(False)
plt.gca().axes.get_yaxis().set_visible(False)
plt.gca().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, diff_embedded)
    dist_embedded[i] = np.inf
    exp_dist_embedded = np.exp(-dist_embedded - logsumexp(-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]), linewidth=5 * thickness[j])


i = 3
relate_point(X, i, plt.gca())

plt.subplot(1, 2, 2)


nca = NeighborhoodComponentsAnalysis(
    max_iter=30,
    random_state=0,
)
nca = nca.fit(X, y)

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

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

plt.gca().set_title("NCA embedding")
plt.gca().axes.get_xaxis().set_visible(False)
plt.gca().axes.get_yaxis().set_visible(False)
plt.gca().axis("equal")

plt.tight_layout()
plt.show()