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# 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

    standard: 0.053 sec
    ltsa: 0.077 sec
    hessian: 0.12 sec
    modified: 0.095 sec
    ISO: 0.18 sec
    MDS: 0.46 sec
    Spectral Embedding: 0.062 sec
    t-SNE: 0.87 sec

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

    from time import time

    import matplotlib.pyplot as plt

    # Unused but required import for doing 3d projections with matplotlib < 3.2
    import mpl_toolkits.mplot3d  # noqa: F401
    import numpy as np
    from matplotlib.ticker import NullFormatter

    from sklearn import manifold
    from sklearn.utils import check_random_state

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

    ax = fig.add_subplot(251, projection="3d")
    ax.scatter(x, y, z, c=p[indices], cmap=plt.cm.rainbow)
    ax.view_init(40, -10)

    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=n_neighbors, n_components=2, method=method, random_state=42
            )
            .fit_transform(sphere_data)
            .T
        )
        t1 = time()
        print("%s: %.2g sec" % (methods[i], t1 - t0))

        ax = fig.add_subplot(252 + i)
        plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
        plt.title("%s (%.2g sec)" % (labels[i], t1 - t0))
        ax.xaxis.set_major_formatter(NullFormatter())
        ax.yaxis.set_major_formatter(NullFormatter())
        plt.axis("tight")

    # Perform Isomap Manifold learning.
    t0 = time()
    trans_data = (
        manifold.Isomap(n_neighbors=n_neighbors, n_components=2)
        .fit_transform(sphere_data)
        .T
    )
    t1 = time()
    print("%s: %.2g sec" % ("ISO", t1 - t0))

    ax = fig.add_subplot(257)
    plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
    plt.title("%s (%.2g sec)" % ("Isomap", t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    plt.axis("tight")

    # Perform Multi-dimensional scaling.
    t0 = time()
    mds = manifold.MDS(2, max_iter=100, n_init=1, normalized_stress="auto", random_state=42)
    trans_data = mds.fit_transform(sphere_data).T
    t1 = time()
    print("MDS: %.2g sec" % (t1 - t0))

    ax = fig.add_subplot(258)
    plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
    plt.title("MDS (%.2g sec)" % (t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    plt.axis("tight")

    # Perform Spectral Embedding.
    t0 = time()
    se = manifold.SpectralEmbedding(
        n_components=2, n_neighbors=n_neighbors, random_state=42
    )
    trans_data = se.fit_transform(sphere_data).T
    t1 = time()
    print("Spectral Embedding: %.2g sec" % (t1 - t0))

    ax = fig.add_subplot(259)
    plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
    plt.title("Spectral Embedding (%.2g sec)" % (t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    plt.axis("tight")

    # Perform t-distributed stochastic neighbor embedding.
    t0 = time()
    tsne = manifold.TSNE(n_components=2, random_state=0)
    trans_data = tsne.fit_transform(sphere_data).T
    t1 = time()
    print("t-SNE: %.2g sec" % (t1 - t0))

    ax = fig.add_subplot(2, 5, 10)
    plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
    plt.title("t-SNE (%.2g sec)" % (t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    plt.axis("tight")

    plt.show()

Total running time of the script: (0 minutes 2.287 seconds)

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