Manifold Learning methods on a severed sphere

An application of the different Manifold learning techniques on a spherical data-set. Here one can see the use of dimensionality reduction in order to gain some intuition regarding the Manifold learning methods. Regarding the dataset, the poles are cut from the sphere, as well as a thin slice down its side. This enables the manifold learning techniques to ‘spread it open’ whilst projecting it onto two dimensions.

For a similar example, where the methods are applied to the S-curve dataset, see Comparison of Manifold Learning methods

Note that the purpose of the MDS is to find a low-dimensional representation of the data (here 2D) in which the distances respect well the distances in the original high-dimensional space, unlike other manifold-learning algorithms, it does not seeks an isotropic representation of the data in the low-dimensional space. Here the manifold problem matches fairly that of representing a flat map of the Earth, as with map projection

Python source code: plot_manifold_sphere.py

# Author: Jaques Grobler <jaques.grobler@inria.fr>
# License: BSD 3 clause

print(__doc__)

from time import time

import numpy as np
import pylab as pl
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.ticker import NullFormatter

from sklearn import manifold
from sklearn.utils import check_random_state

# Next line to silence pyflakes.
Axes3D

# Variables for manifold learning.
n_neighbors = 10
n_samples = 1000

# Create our sphere.
random_state = check_random_state(0)
p = random_state.rand(n_samples) * (2 * np.pi - 0.55)
t = random_state.rand(n_samples) * np.pi

# Sever the poles from the sphere.
indices = ((t < (np.pi - (np.pi / 8))) & (t > ((np.pi / 8))))
colors = p[indices]
x, y, z = np.sin(t[indices]) * np.cos(p[indices]), \
    np.sin(t[indices]) * np.sin(p[indices]), \
    np.cos(t[indices])

# Plot our dataset.
fig = pl.figure(figsize=(15, 8))
pl.suptitle("Manifold Learning with %i points, %i neighbors"
            % (1000, n_neighbors), fontsize=14)

ax = fig.add_subplot(241, projection='3d')
ax.scatter(x, y, z, c=p[indices], cmap=pl.cm.rainbow)
try:
    # compatibility matplotlib < 1.0
    ax.view_init(40, -10)
except:
    pass

sphere_data = np.array([x, y, z]).T

# Perform Locally Linear Embedding Manifold learning
methods = ['standard', 'ltsa', 'hessian', 'modified']
labels = ['LLE', 'LTSA', 'Hessian LLE', 'Modified LLE']

for i, method in enumerate(methods):
    t0 = time()
    trans_data = manifold\
        .LocallyLinearEmbedding(n_neighbors, 2,
                                method=method).fit_transform(sphere_data).T
    t1 = time()
    print("%s: %.2g sec" % (methods[i], t1 - t0))

    ax = fig.add_subplot(242 + i)
    pl.scatter(trans_data[0], trans_data[1], c=colors, cmap=pl.cm.rainbow)
    pl.title("%s (%.2g sec)" % (labels[i], t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    pl.axis('tight')

# Perform Isomap Manifold learning.
t0 = time()
trans_data = manifold.Isomap(n_neighbors, n_components=2)\
    .fit_transform(sphere_data).T
t1 = time()
print("%s: %.2g sec" % ('ISO', t1 - t0))

ax = fig.add_subplot(246)
pl.scatter(trans_data[0], trans_data[1],  c=colors, cmap=pl.cm.rainbow)
pl.title("%s (%.2g sec)" % ('Isomap', t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
pl.axis('tight')

# Perform Multi-dimensional scaling.
t0 = time()
mds = manifold.MDS(2, max_iter=100, n_init=1)
trans_data = mds.fit_transform(sphere_data).T
t1 = time()
print("MDS: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(247)
pl.scatter(trans_data[0], trans_data[1],  c=colors, cmap=pl.cm.rainbow)
pl.title("MDS (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
pl.axis('tight')

# Perform Spectral Embedding.
t0 = time()
se = manifold.SpectralEmbedding(n_components=2,
                                n_neighbors=n_neighbors)
trans_data = se.fit_transform(sphere_data).T
t1 = time()
print("Spectral Embedding: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(248)
pl.scatter(trans_data[0], trans_data[1],  c=colors, cmap=pl.cm.rainbow)
pl.title("Spectral Embedding (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
pl.axis('tight')

pl.show()