scikit-learn: Comparison of the K-Means and MiniBatchKMeans clustering algorithms

I am going through the scikit-learn user guide on Clustering. They have an example comparing K-Means and MiniBatchKMeans.

I am a little confused about the following code in the example:

# We want to have the same colors for the same cluster from the
# MiniBatchKMeans and the KMeans algorithm. Let's pair the cluster centers per
# closest one.
k_means_cluster_centers = np.sort(k_means.cluster_centers_, axis=0)
mbk_means_cluster_centers = np.sort(mbk.cluster_centers_, axis=0)
k_means_labels = pairwise_distances_argmin(X, k_means_cluster_centers)
mbk_means_labels = pairwise_distances_argmin(X, mbk_means_cluster_centers)
order = pairwise_distances_argmin(k_means_cluster_centers,
                                  mbk_means_cluster_centers)

The values of the k-means cluster centers before and after sorting are:

k_means.cluster_centers_
array([[ 1.07705469, -1.06730994],
       [-1.07159013, -1.00648645],
       [ 0.96700708,  1.01837274]])

k_means_cluster_centers
array([[-1.07159013, -1.06730994],
       [ 0.96700708, -1.00648645],
       [ 1.07705469,  1.01837274]])

There are three centers, so I suppose each row is the xy-coordinate of one center. I am not sure why they use np.sort() before pairing each point with the closest center because this distorts the x/y coordinates of the centers. Maybe they were trying to sort just by x or y axis?

Solution

I think you are right. Sorting like it is done in this example mixes up the x and y coordinates of the points. The fact that it works in the example is more or less coincidence.

We have x-coordinates [1, -1, 1] and y-coordinates [1, -1, -1]. Sorted they become [-1, 1, 1] and [-1, -1, 1] which form the same three pairs we had originally:

# original | sorted
# [ 1, -1] | [-1, -1]
# [-1, -1] | [ 1, -1]
# [ 1,  1] | [ 1,  1]

Observe below how this breaks down when using four clusters. In this case we have:

# original | sorted
# [-1, -1] | [-1, -1]
# [-1,  1] | [-1, -1]
# [ 1, -1] | [ 1,  1]
# [ 1,  1] | [ 1,  1]

which are not the same points.

Modified example code:

print(__doc__)

import time

import numpy as np
import matplotlib.pyplot as plt

from sklearn.cluster import MiniBatchKMeans, KMeans
from sklearn.metrics.pairwise import pairwise_distances_argmin
from sklearn.datasets.samples_generator import make_blobs

# #############################################################################
# Generate sample data
np.random.seed(0)

batch_size = 45
centers = [[1, 1], [-1, -1], [1, -1], [-1, 1]]
n_clusters = len(centers)
X, labels_true = make_blobs(n_samples=3000, centers=centers, cluster_std=0.7)

# #############################################################################
# Compute clustering with Means

k_means = KMeans(init='k-means++', n_clusters=4, n_init=10)
t0 = time.time()
k_means.fit(X)
t_batch = time.time() - t0

# #############################################################################
# Compute clustering with MiniBatchKMeans

mbk = MiniBatchKMeans(init='k-means++', n_clusters=4, batch_size=batch_size,
                      n_init=10, max_no_improvement=10, verbose=0)
t0 = time.time()
mbk.fit(X)
t_mini_batch = time.time() - t0

# #############################################################################
# Plot result

fig = plt.figure(figsize=(8, 3))
fig.subplots_adjust(left=0.02, right=0.98, bottom=0.05, top=0.9)
colors = ['#4EACC5', '#FF9C34', '#4E9A06', '#123456']

# We want to have the same colors for the same cluster from the
# MiniBatchKMeans and the KMeans algorithm. Let's pair the cluster centers per
# closest one.
k_means_cluster_centers = np.sort(k_means.cluster_centers_, axis=0)
mbk_means_cluster_centers = np.sort(mbk.cluster_centers_, axis=0)
k_means_labels = pairwise_distances_argmin(X, k_means_cluster_centers)
mbk_means_labels = pairwise_distances_argmin(X, mbk_means_cluster_centers)
order = pairwise_distances_argmin(k_means_cluster_centers,
                                  mbk_means_cluster_centers)

# KMeans
ax = fig.add_subplot(1, 3, 1)
for k, col in zip(range(n_clusters), colors):
    my_members = k_means_labels == k
    cluster_center = k_means_cluster_centers[k]
    ax.plot(X[my_members, 0], X[my_members, 1], 'w',
            markerfacecolor=col, marker='.')
    ax.plot(cluster_center[0], cluster_center[1], 'o', markerfacecolor=col,
            markeredgecolor='k', markersize=6)
ax.set_title('KMeans')
ax.set_xticks(())
ax.set_yticks(())
plt.text(-3.5, 1.8,  'train time: %.2fs\ninertia: %f' % (
    t_batch, k_means.inertia_))

# MiniBatchKMeans
ax = fig.add_subplot(1, 3, 2)
for k, col in zip(range(n_clusters), colors):
    my_members = mbk_means_labels == order[k]
    cluster_center = mbk_means_cluster_centers[order[k]]
    ax.plot(X[my_members, 0], X[my_members, 1], 'w',
            markerfacecolor=col, marker='.')
    ax.plot(cluster_center[0], cluster_center[1], 'o', markerfacecolor=col,
            markeredgecolor='k', markersize=6)
ax.set_title('MiniBatchKMeans')
ax.set_xticks(())
ax.set_yticks(())
plt.text(-3.5, 1.8, 'train time: %.2fs\ninertia: %f' %
         (t_mini_batch, mbk.inertia_))

# Initialise the different array to all False
different = (mbk_means_labels == 4)
ax = fig.add_subplot(1, 3, 3)

for k in range(n_clusters):
    different += ((k_means_labels == k) != (mbk_means_labels == order[k]))

identic = np.logical_not(different)
ax.plot(X[identic, 0], X[identic, 1], 'w',
        markerfacecolor='#bbbbbb', marker='.')
ax.plot(X[different, 0], X[different, 1], 'w',
        markerfacecolor='m', marker='.')
ax.set_title('Difference')
ax.set_xticks(())
ax.set_yticks(())

plt.show()

A more appropriate ordering might look like this:

# order cluster centers by their x and y coordinates, weighted by 1 and 0.1 respectively
k_order = np.argsort(k_means.cluster_centers_[:, 0] + k_means.cluster_centers_[:, 1]*0.1)
mbk_order = np.argsort(mbk.cluster_centers_[:, 0] + mbk.cluster_centers_[:, 1]*0.1)
k_means_cluster_centers = k_means.cluster_centers_[k_order]
mbk_means_cluster_centers = mbk.cluster_centers_[mbk_order]

However, the correct approach would be to first align the cluster centers, and then impose an (arbitrary) order. This should do the job:

mbk_order = pairwise_distances_argmin(k_means.cluster_centers_, mbk.cluster_centers_)
k_means_cluster_centers = k_means.cluster_centers_
mbk_means_cluster_centers = mbk.cluster_centers_[mbk_order]