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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_manifold_plot_swissroll.py>`
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.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_manifold_plot_swissroll.py:


===================================
Swiss Roll And Swiss-Hole Reduction
===================================
This notebook seeks to compare two popular non-linear dimensionality
techniques, T-distributed Stochastic Neighbor Embedding (t-SNE) and
Locally Linear Embedding (LLE), on the classic Swiss Roll dataset.
Then, we will explore how they both deal with the addition of a hole
in the data.

.. GENERATED FROM PYTHON SOURCE LINES 11-15

.. code-block:: Python


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








.. GENERATED FROM PYTHON SOURCE LINES 16-20

Swiss Roll
---------------------------------------------------

We start by generating the Swiss Roll dataset.

.. GENERATED FROM PYTHON SOURCE LINES 20-27

.. code-block:: Python


    import matplotlib.pyplot as plt

    from sklearn import datasets, manifold

    sr_points, sr_color = datasets.make_swiss_roll(n_samples=1500, random_state=0)








.. GENERATED FROM PYTHON SOURCE LINES 28-29

Now, let's take a look at our data:

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    fig = plt.figure(figsize=(8, 6))
    ax = fig.add_subplot(111, projection="3d")
    fig.add_axes(ax)
    ax.scatter(
        sr_points[:, 0], sr_points[:, 1], sr_points[:, 2], c=sr_color, s=50, alpha=0.8
    )
    ax.set_title("Swiss Roll in Ambient Space")
    ax.view_init(azim=-66, elev=12)
    _ = ax.text2D(0.8, 0.05, s="n_samples=1500", transform=ax.transAxes)




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_swissroll_001.png
   :alt: Swiss Roll in Ambient Space
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_swissroll_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-46

Computing the LLE and t-SNE embeddings, we find that LLE seems to unroll the
Swiss Roll pretty effectively. t-SNE on the other hand, is able
to preserve the general structure of the data, but, poorly represents the
continuous nature of our original data. Instead, it seems to unnecessarily
clump sections of points together.

.. GENERATED FROM PYTHON SOURCE LINES 46-61

.. code-block:: Python


    sr_lle, sr_err = manifold.locally_linear_embedding(
        sr_points, n_neighbors=12, n_components=2
    )

    sr_tsne = manifold.TSNE(n_components=2, perplexity=40, random_state=0).fit_transform(
        sr_points
    )

    fig, axs = plt.subplots(figsize=(8, 8), nrows=2)
    axs[0].scatter(sr_lle[:, 0], sr_lle[:, 1], c=sr_color)
    axs[0].set_title("LLE Embedding of Swiss Roll")
    axs[1].scatter(sr_tsne[:, 0], sr_tsne[:, 1], c=sr_color)
    _ = axs[1].set_title("t-SNE Embedding of Swiss Roll")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_swissroll_002.png
   :alt: LLE Embedding of Swiss Roll, t-SNE Embedding of Swiss Roll
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_swissroll_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 62-69

.. note::

    LLE seems to be stretching the points from the center (purple)
    of the swiss roll. However, we observe that this is simply a byproduct
    of how the data was generated. There is a higher density of points near the
    center of the roll, which ultimately affects how LLE reconstructs the
    data in a lower dimension.

.. GENERATED FROM PYTHON SOURCE LINES 71-76

Swiss-Hole
---------------------------------------------------

Now let's take a look at how both algorithms deal with us adding a hole to
the data. First, we generate the Swiss-Hole dataset and plot it:

.. GENERATED FROM PYTHON SOURCE LINES 76-91

.. code-block:: Python


    sh_points, sh_color = datasets.make_swiss_roll(
        n_samples=1500, hole=True, random_state=0
    )

    fig = plt.figure(figsize=(8, 6))
    ax = fig.add_subplot(111, projection="3d")
    fig.add_axes(ax)
    ax.scatter(
        sh_points[:, 0], sh_points[:, 1], sh_points[:, 2], c=sh_color, s=50, alpha=0.8
    )
    ax.set_title("Swiss-Hole in Ambient Space")
    ax.view_init(azim=-66, elev=12)
    _ = ax.text2D(0.8, 0.05, s="n_samples=1500", transform=ax.transAxes)




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_swissroll_003.png
   :alt: Swiss-Hole in Ambient Space
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_swissroll_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 92-96

Computing the LLE and t-SNE embeddings, we obtain similar results to the
Swiss Roll. LLE very capably unrolls the data and even preserves
the hole. t-SNE, again seems to clump sections of points together, but, we
note that it preserves the general topology of the original data.

.. GENERATED FROM PYTHON SOURCE LINES 96-112

.. code-block:: Python



    sh_lle, sh_err = manifold.locally_linear_embedding(
        sh_points, n_neighbors=12, n_components=2
    )

    sh_tsne = manifold.TSNE(
        n_components=2, perplexity=40, init="random", random_state=0
    ).fit_transform(sh_points)

    fig, axs = plt.subplots(figsize=(8, 8), nrows=2)
    axs[0].scatter(sh_lle[:, 0], sh_lle[:, 1], c=sh_color)
    axs[0].set_title("LLE Embedding of Swiss-Hole")
    axs[1].scatter(sh_tsne[:, 0], sh_tsne[:, 1], c=sh_color)
    _ = axs[1].set_title("t-SNE Embedding of Swiss-Hole")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_swissroll_004.png
   :alt: LLE Embedding of Swiss-Hole, t-SNE Embedding of Swiss-Hole
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_swissroll_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 113-123

Concluding remarks
------------------

We note that t-SNE benefits from testing more combinations of parameters.
Better results could probably have been obtained by better tuning these
parameters.

We observe that, as seen in the "Manifold learning on
handwritten digits" example, t-SNE generally performs better than LLE
on real world data.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 9.040 seconds)


.. _sphx_glr_download_auto_examples_manifold_plot_swissroll.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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