Note

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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.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

Dataset preparation#

We start by generating the S-curve dataset.

import matplotlib.pyplot as plt

# unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d  # noqa: F401
from matplotlib import ticker

from sklearn import datasets, manifold

n_samples = 1500
S_points, S_color = datasets.make_s_curve(n_samples, random_state=0)

Let’s look at the original data. Also define some helping functions, which we will use further on.

def plot_3d(points, points_color, title):
    x, y, z = points.T

    fig, ax = plt.subplots(
        figsize=(6, 6),
        facecolor="white",
        tight_layout=True,
        subplot_kw={"projection": "3d"},
    )
    fig.suptitle(title, size=16)
    col = ax.scatter(x, y, z, c=points_color, s=50, alpha=0.8)
    ax.view_init(azim=-60, elev=9)
    ax.xaxis.set_major_locator(ticker.MultipleLocator(1))
    ax.yaxis.set_major_locator(ticker.MultipleLocator(1))
    ax.zaxis.set_major_locator(ticker.MultipleLocator(1))

    fig.colorbar(col, ax=ax, orientation="horizontal", shrink=0.6, aspect=60, pad=0.01)
    plt.show()


def plot_2d(points, points_color, title):
    fig, ax = plt.subplots(figsize=(3, 3), facecolor="white", constrained_layout=True)
    fig.suptitle(title, size=16)
    add_2d_scatter(ax, points, points_color)
    plt.show()


def add_2d_scatter(ax, points, points_color, title=None):
    x, y = points.T
    ax.scatter(x, y, c=points_color, s=50, alpha=0.8)
    ax.set_title(title)
    ax.xaxis.set_major_formatter(ticker.NullFormatter())
    ax.yaxis.set_major_formatter(ticker.NullFormatter())


plot_3d(S_points, S_color, "Original S-curve samples")

Define algorithms for the manifold learning#

Manifold learning is an approach to non-linear dimensionality reduction. Algorithms for this task are based on the idea that the dimensionality of many data sets is only artificially high.

Locally Linear Embeddings#

Locally linear embedding (LLE) can be thought of as a series of local Principal Component Analyses which are globally compared to find the best non-linear embedding. Read more in the User Guide.

params = {
    "n_neighbors": n_neighbors,
    "n_components": n_components,
    "eigen_solver": "auto",
    "random_state": 0,
}

lle_standard = manifold.LocallyLinearEmbedding(method="standard", **params)
S_standard = lle_standard.fit_transform(S_points)

lle_ltsa = manifold.LocallyLinearEmbedding(method="ltsa", **params)
S_ltsa = lle_ltsa.fit_transform(S_points)

lle_hessian = manifold.LocallyLinearEmbedding(method="hessian", **params)
S_hessian = lle_hessian.fit_transform(S_points)

lle_mod = manifold.LocallyLinearEmbedding(method="modified", **params)
S_mod = lle_mod.fit_transform(S_points)

fig, axs = plt.subplots(
    nrows=2, ncols=2, figsize=(7, 7), facecolor="white", constrained_layout=True
)
fig.suptitle("Locally Linear Embeddings", size=16)

lle_methods = [
    ("Standard locally linear embedding", S_standard),
    ("Local tangent space alignment", S_ltsa),
    ("Hessian eigenmap", S_hessian),
    ("Modified locally linear embedding", S_mod),
]
for ax, method in zip(axs.flat, lle_methods):
    name, points = method
    add_2d_scatter(ax, points, S_color, name)

plt.show()

Locally Linear Embeddings, Standard locally linear embedding, Local tangent space alignment, Hessian eigenmap, Modified locally linear embedding

Isomap Embedding#

Non-linear dimensionality reduction through Isometric Mapping. Isomap seeks a lower-dimensional embedding which maintains geodesic distances between all points. Read more in the User Guide.

isomap = manifold.Isomap(n_neighbors=n_neighbors, n_components=n_components, p=1)
S_isomap = isomap.fit_transform(S_points)

plot_2d(S_isomap, S_color, "Isomap Embedding")

Multidimensional scaling#

Multidimensional scaling (MDS) seeks a low-dimensional representation of the data in which the distances respect well the distances in the original high-dimensional space. Read more in the User Guide.

md_scaling = manifold.MDS(
    n_components=n_components,
    max_iter=50,
    n_init=4,
    random_state=0,
    normalized_stress=False,
)
S_scaling = md_scaling.fit_transform(S_points)

plot_2d(S_scaling, S_color, "Multidimensional scaling")

Spectral embedding for non-linear dimensionality reduction#

This implementation uses Laplacian Eigenmaps, which finds a low dimensional representation of the data using a spectral decomposition of the graph Laplacian. Read more in the User Guide.

spectral = manifold.SpectralEmbedding(
    n_components=n_components, n_neighbors=n_neighbors, random_state=42
)
S_spectral = spectral.fit_transform(S_points)

plot_2d(S_spectral, S_color, "Spectral Embedding")

T-distributed Stochastic Neighbor Embedding#

It converts similarities between data points to joint probabilities and tries to minimize the Kullback-Leibler divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data. t-SNE has a cost function that is not convex, i.e. with different initializations we can get different results. Read more in the User Guide.

t_sne = manifold.TSNE(
    n_components=n_components,
    perplexity=30,
    init="random",
    max_iter=250,
    random_state=0,
)
S_t_sne = t_sne.fit_transform(S_points)

plot_2d(S_t_sne, S_color, "T-distributed Stochastic  \n Neighbor Embedding")