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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/manifold/plot_swissroll.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_examples_manifold_plot_swissroll.py>`
        to download the full example code or to run this example in your browser via Binder

.. 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 12-16

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

We start by generating the Swiss Roll dataset.

.. GENERATED FROM PYTHON SOURCE LINES 16-23

.. code-block:: default


    import matplotlib.pyplot as plt
    from sklearn import manifold, datasets


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








.. GENERATED FROM PYTHON SOURCE LINES 24-25

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

.. GENERATED FROM PYTHON SOURCE LINES 25-36

.. code-block:: default


    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 37-42

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 42-57

.. code-block:: default


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

    sr_tsne = manifold.TSNE(
        n_components=2, learning_rate="auto", perplexity=40, init="pca", 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


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    /home/runner/work/scikit-learn/scikit-learn/sklearn/manifold/_t_sne.py:996: FutureWarning: The PCA initialization in TSNE will change to have the standard deviation of PC1 equal to 1e-4 in 1.2. This will ensure better convergence.
      warnings.warn(




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.. 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 67-72

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 72-87

.. code-block:: default


    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 88-92

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 92-108

.. code-block:: default



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

    sh_tsne = manifold.TSNE(
        n_components=2, learning_rate="auto", 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 109-119

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  18.006 seconds)


.. _sphx_glr_download_auto_examples_manifold_plot_swissroll.py:

.. only:: html

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


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