Comparison of Manifold Learning methods

An illustration of dimensionality reduction on the S-curve dataset with various manifold learning methods.

For a discussion and comparison of these algorithms, see the manifold module page

For a similar example, where the methods are applied to a sphere dataset, see Manifold Learning methods on a severed sphere

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.

Manifold Learning with 1000 points, 10 neighbors, LLE (0.095 sec), LTSA (0.19 sec), Hessian LLE (0.28 sec), Modified LLE (0.2 sec), Isomap (0.39 sec), MDS (2.2 sec), SE (0.09 sec), t-SNE (8.2 sec)


/home/circleci/project/sklearn/utils/ FutureWarning: Pass n_neighbors=10, n_components=2 as keyword args. From version 0.25 passing these as positional arguments will result in an error
  warnings.warn("Pass {} as keyword args. From version 0.25 "
LLE: 0.095 sec
LTSA: 0.19 sec
Hessian LLE: 0.28 sec
Modified LLE: 0.2 sec
Isomap: 0.39 sec
MDS: 2.2 sec
SE: 0.09 sec
t-SNE: 8.2 sec

# Author: Jake Vanderplas -- <>


from collections import OrderedDict
from functools import partial
from time import time

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.ticker import NullFormatter

from sklearn import manifold, datasets

# Next line to silence pyflakes. This import is needed.

n_points = 1000
X, color = datasets.make_s_curve(n_points, random_state=0)
n_neighbors = 10
n_components = 2

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

# Add 3d scatter plot
ax = fig.add_subplot(251, projection='3d')
ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=color,
ax.view_init(4, -72)

# Set-up manifold methods
LLE = partial(manifold.LocallyLinearEmbedding,
              n_neighbors, n_components, eigen_solver='auto')

methods = OrderedDict()
methods['LLE'] = LLE(method='standard')
methods['LTSA'] = LLE(method='ltsa')
methods['Hessian LLE'] = LLE(method='hessian')
methods['Modified LLE'] = LLE(method='modified')
methods['Isomap'] = manifold.Isomap(n_neighbors, n_components)
methods['MDS'] = manifold.MDS(n_components, max_iter=100, n_init=1)
methods['SE'] = manifold.SpectralEmbedding(n_components=n_components,
methods['t-SNE'] = manifold.TSNE(n_components=n_components, init='pca',

# Plot results
for i, (label, method) in enumerate(methods.items()):
    t0 = time()
    Y = method.fit_transform(X)
    t1 = time()
    print("%s: %.2g sec" % (label, t1 - t0))
    ax = fig.add_subplot(2, 5, 2 + i + (i > 3))
    ax.scatter(Y[:, 0], Y[:, 1], c=color,
    ax.set_title("%s (%.2g sec)" % (label, t1 - t0))

Total running time of the script: ( 0 minutes 12.202 seconds)

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