
.. currentmodule:: sklearn.manifold

.. _manifold:

=================
Manifold learning
=================

.. rst-class:: quote

                 | Look for the bare necessities
                 | The simple bare necessities
                 | Forget about your worries and your strife
                 | I mean the bare necessities
                 | Old Mother Nature's recipes
                 | That bring the bare necessities of life
                 |
                 |             -- Baloo's song [The Jungle Book]



.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_compare_methods_001.png
   :target: ../auto_examples/manifold/plot_compare_methods.html
   :align: center
   :scale: 60

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.


Introduction
============

High-dimensional datasets can be very difficult to visualize.  While data
in two or three dimensions can be plotted to show the inherent
structure of the data, equivalent high-dimensional plots are much less
intuitive.  To aid visualization of the structure of a dataset, the
dimension must be reduced in some way.

The simplest way to accomplish this dimensionality reduction is by taking
a random projection of the data.  Though this allows some degree of
visualization of the data structure, the randomness of the choice leaves much
to be desired.  In a random projection, it is likely that the more
interesting structure within the data will be lost.


.. |digits_img| image:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_001.png
    :target: ../auto_examples/manifold/plot_lle_digits.html
    :scale: 50

.. |projected_img| image::  ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_002.png
    :target: ../auto_examples/manifold/plot_lle_digits.html
    :scale: 50

.. centered:: |digits_img| |projected_img|


To address this concern, a number of supervised and unsupervised linear
dimensionality reduction frameworks have been designed, such as Principal
Component Analysis (PCA), Independent Component Analysis, Linear
Discriminant Analysis, and others.  These algorithms define specific
rubrics to choose an "interesting" linear projection of the data.
These methods can be powerful, but often miss important non-linear
structure in the data.


.. |PCA_img| image:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_003.png
    :target: ../auto_examples/manifold/plot_lle_digits.html
    :scale: 50

.. |LDA_img| image::  ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_004.png
    :target: ../auto_examples/manifold/plot_lle_digits.html
    :scale: 50

.. centered:: |PCA_img| |LDA_img|

Manifold Learning can be thought of as an attempt to generalize linear
frameworks like PCA to be sensitive to non-linear structure in data. Though
supervised variants exist, the typical manifold learning problem is
unsupervised: it learns the high-dimensional structure of the data
from the data itself, without the use of predetermined classifications.


.. topic:: Examples:

    * See :ref:`sphx_glr_auto_examples_manifold_plot_lle_digits.py` for an example of
      dimensionality reduction on handwritten digits.

    * See :ref:`sphx_glr_auto_examples_manifold_plot_compare_methods.py` for an example of
      dimensionality reduction on a toy "S-curve" dataset.

The manifold learning implementations available in scikit-learn are
summarized below

.. _isomap:

Isomap
======

One of the earliest approaches to manifold learning is the Isomap
algorithm, short for Isometric Mapping.  Isomap can be viewed as an
extension of Multi-dimensional Scaling (MDS) or Kernel PCA.
Isomap seeks a lower-dimensional embedding which maintains geodesic
distances between all points.  Isomap can be performed with the object
:class:`Isomap`.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_005.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Complexity
----------
The Isomap algorithm comprises three stages:

1. **Nearest neighbor search.**  Isomap uses
   :class:`sklearn.neighbors.BallTree` for efficient neighbor search.
   The cost is approximately :math:`O[D \log(k) N \log(N)]`, for :math:`k`
   nearest neighbors of :math:`N` points in :math:`D` dimensions.

2. **Shortest-path graph search.**  The most efficient known algorithms
   for this are *Dijkstra's Algorithm*, which is approximately
   :math:`O[N^2(k + \log(N))]`, or the *Floyd-Warshall algorithm*, which
   is :math:`O[N^3]`.  The algorithm can be selected by the user with
   the ``path_method`` keyword of ``Isomap``.  If unspecified, the code
   attempts to choose the best algorithm for the input data.

3. **Partial eigenvalue decomposition.**  The embedding is encoded in the
   eigenvectors corresponding to the :math:`d` largest eigenvalues of the
   :math:`N \times N` isomap kernel.  For a dense solver, the cost is
   approximately :math:`O[d N^2]`.  This cost can often be improved using
   the ``ARPACK`` solver.  The eigensolver can be specified by the user
   with the ``path_method`` keyword of ``Isomap``.  If unspecified, the
   code attempts to choose the best algorithm for the input data.

The overall complexity of Isomap is
:math:`O[D \log(k) N \log(N)] + O[N^2(k + \log(N))] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"A global geometric framework for nonlinear dimensionality reduction"
     <http://science.sciencemag.org/content/290/5500/2319.full>`_
     Tenenbaum, J.B.; De Silva, V.; & Langford, J.C.  Science 290 (5500)

.. _locally_linear_embedding:

Locally Linear Embedding
========================

Locally linear embedding (LLE) seeks a lower-dimensional projection of the data
which preserves distances within local neighborhoods.  It can be thought
of as a series of local Principal Component Analyses which are globally
compared to find the best non-linear embedding.

Locally linear embedding can be performed with function
:func:`locally_linear_embedding` or its object-oriented counterpart
:class:`LocallyLinearEmbedding`.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_006.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Complexity
----------

The standard LLE algorithm comprises three stages:

1. **Nearest Neighbors Search**.  See discussion under Isomap above.

2. **Weight Matrix Construction**. :math:`O[D N k^3]`.
   The construction of the LLE weight matrix involves the solution of a
   :math:`k \times k` linear equation for each of the :math:`N` local
   neighborhoods

3. **Partial Eigenvalue Decomposition**. See discussion under Isomap above.

The overall complexity of standard LLE is
:math:`O[D \log(k) N \log(N)] + O[D N k^3] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"Nonlinear dimensionality reduction by locally linear embedding"
     <http://www.sciencemag.org/content/290/5500/2323.full>`_
     Roweis, S. & Saul, L.  Science 290:2323 (2000)


Modified Locally Linear Embedding
=================================

One well-known issue with LLE is the regularization problem.  When the number
of neighbors is greater than the number of input dimensions, the matrix
defining each local neighborhood is rank-deficient.  To address this, standard
LLE applies an arbitrary regularization parameter :math:`r`, which is chosen
relative to the trace of the local weight matrix.  Though it can be shown
formally that as :math:`r \to 0`, the solution converges to the desired
embedding, there is no guarantee that the optimal solution will be found
for :math:`r > 0`.  This problem manifests itself in embeddings which distort
the underlying geometry of the manifold.

One method to address the regularization problem is to use multiple weight
vectors in each neighborhood.  This is the essence of *modified locally
linear embedding* (MLLE).  MLLE can be  performed with function
:func:`locally_linear_embedding` or its object-oriented counterpart
:class:`LocallyLinearEmbedding`, with the keyword ``method = 'modified'``.
It requires ``n_neighbors > n_components``.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_007.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Complexity
----------

The MLLE algorithm comprises three stages:

1. **Nearest Neighbors Search**.  Same as standard LLE

2. **Weight Matrix Construction**. Approximately
   :math:`O[D N k^3] + O[N (k-D) k^2]`.  The first term is exactly equivalent
   to that of standard LLE.  The second term has to do with constructing the
   weight matrix from multiple weights.  In practice, the added cost of
   constructing the MLLE weight matrix is relatively small compared to the
   cost of stages 1 and 3.

3. **Partial Eigenvalue Decomposition**. Same as standard LLE

The overall complexity of MLLE is
:math:`O[D \log(k) N \log(N)] + O[D N k^3] + O[N (k-D) k^2] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"MLLE: Modified Locally Linear Embedding Using Multiple Weights"
     <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.382>`_
     Zhang, Z. & Wang, J.


Hessian Eigenmapping
====================

Hessian Eigenmapping (also known as Hessian-based LLE: HLLE) is another method
of solving the regularization problem of LLE.  It revolves around a
hessian-based quadratic form at each neighborhood which is used to recover
the locally linear structure.  Though other implementations note its poor
scaling with data size, ``sklearn`` implements some algorithmic
improvements which make its cost comparable to that of other LLE variants
for small output dimension.  HLLE can be  performed with function
:func:`locally_linear_embedding` or its object-oriented counterpart
:class:`LocallyLinearEmbedding`, with the keyword ``method = 'hessian'``.
It requires ``n_neighbors > n_components * (n_components + 3) / 2``.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_008.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Complexity
----------

The HLLE algorithm comprises three stages:

1. **Nearest Neighbors Search**.  Same as standard LLE

2. **Weight Matrix Construction**. Approximately
   :math:`O[D N k^3] + O[N d^6]`.  The first term reflects a similar
   cost to that of standard LLE.  The second term comes from a QR
   decomposition of the local hessian estimator.

3. **Partial Eigenvalue Decomposition**. Same as standard LLE

The overall complexity of standard HLLE is
:math:`O[D \log(k) N \log(N)] + O[D N k^3] + O[N d^6] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"Hessian Eigenmaps: Locally linear embedding techniques for
     high-dimensional data" <http://www.pnas.org/content/100/10/5591>`_
     Donoho, D. & Grimes, C. Proc Natl Acad Sci USA. 100:5591 (2003)

.. _spectral_embedding:

Spectral Embedding
====================

Spectral Embedding is an approach to calculating a non-linear embedding.
Scikit-learn implements Laplacian Eigenmaps, which finds a low dimensional
representation of the data using a spectral decomposition of the graph
Laplacian. The graph generated can be considered as a discrete approximation of
the low dimensional manifold in the high dimensional space. Minimization of a
cost function based on the graph ensures that points close to each other on
the manifold are mapped close to each other in the low dimensional space,
preserving local distances. Spectral embedding can be  performed with the
function :func:`spectral_embedding` or its object-oriented counterpart
:class:`SpectralEmbedding`.

Complexity
----------

The Spectral Embedding (Laplacian Eigenmaps) algorithm comprises three stages:

1. **Weighted Graph Construction**. Transform the raw input data into
   graph representation using affinity (adjacency) matrix representation.

2. **Graph Laplacian Construction**. unnormalized Graph Laplacian
   is constructed as :math:`L = D - A` for and normalized one as
   :math:`L = D^{-\frac{1}{2}} (D - A) D^{-\frac{1}{2}}`.

3. **Partial Eigenvalue Decomposition**. Eigenvalue decomposition is
   done on graph Laplacian

The overall complexity of spectral embedding is
:math:`O[D \log(k) N \log(N)] + O[D N k^3] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"Laplacian Eigenmaps for Dimensionality Reduction
     and Data Representation"
     <https://web.cse.ohio-state.edu/~mbelkin/papers/LEM_NC_03.pdf>`_
     M. Belkin, P. Niyogi, Neural Computation, June 2003; 15 (6):1373-1396


Local Tangent Space Alignment
=============================

Though not technically a variant of LLE, Local tangent space alignment (LTSA)
is algorithmically similar enough to LLE that it can be put in this category.
Rather than focusing on preserving neighborhood distances as in LLE, LTSA
seeks to characterize the local geometry at each neighborhood via its
tangent space, and performs a global optimization to align these local
tangent spaces to learn the embedding.  LTSA can be performed with function
:func:`locally_linear_embedding` or its object-oriented counterpart
:class:`LocallyLinearEmbedding`, with the keyword ``method = 'ltsa'``.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_009.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Complexity
----------

The LTSA algorithm comprises three stages:

1. **Nearest Neighbors Search**.  Same as standard LLE

2. **Weight Matrix Construction**. Approximately
   :math:`O[D N k^3] + O[k^2 d]`.  The first term reflects a similar
   cost to that of standard LLE.

3. **Partial Eigenvalue Decomposition**. Same as standard LLE

The overall complexity of standard LTSA is
:math:`O[D \log(k) N \log(N)] + O[D N k^3] + O[k^2 d] + O[d N^2]`.

* :math:`N` : number of training data points
* :math:`D` : input dimension
* :math:`k` : number of nearest neighbors
* :math:`d` : output dimension

.. topic:: References:

   * `"Principal manifolds and nonlinear dimensionality reduction via
     tangent space alignment"
     <http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.3693>`_
     Zhang, Z. & Zha, H. Journal of Shanghai Univ. 8:406 (2004)

.. _multidimensional_scaling:

Multi-dimensional Scaling (MDS)
===============================

`Multidimensional scaling <https://en.wikipedia.org/wiki/Multidimensional_scaling>`_
(:class:`MDS`) seeks a low-dimensional
representation of the data in which the distances respect well the
distances in the original high-dimensional space.

In general, :class:`MDS` is a technique used for analyzing similarity or
dissimilarity data. It attempts to model similarity or dissimilarity data as
distances in a geometric spaces. The data can be ratings of similarity between
objects, interaction frequencies of molecules, or trade indices between
countries.

There exists two types of MDS algorithm: metric and non metric. In the
scikit-learn, the class :class:`MDS` implements both. In Metric MDS, the input
similarity matrix arises from a metric (and thus respects the triangular
inequality), the distances between output two points are then set to be as
close as possible to the similarity or dissimilarity data. In the non-metric
version, the algorithms will try to preserve the order of the distances, and
hence seek for a monotonic relationship between the distances in the embedded
space and the similarities/dissimilarities.

.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_010.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50


Let :math:`S` be the similarity matrix, and :math:`X` the coordinates of the
:math:`n` input points. Disparities :math:`\hat{d}_{ij}` are transformation of
the similarities chosen in some optimal ways. The objective, called the
stress, is then defined by :math:`\sum_{i < j} d_{ij}(X) - \hat{d}_{ij}(X)`


Metric MDS
----------

The simplest metric :class:`MDS` model, called *absolute MDS*, disparities are defined by
:math:`\hat{d}_{ij} = S_{ij}`. With absolute MDS, the value :math:`S_{ij}`
should then correspond exactly to the distance between point :math:`i` and
:math:`j` in the embedding point.

Most commonly, disparities are set to :math:`\hat{d}_{ij} = b S_{ij}`.

Nonmetric MDS
-------------

Non metric :class:`MDS` focuses on the ordination of the data. If
:math:`S_{ij} < S_{jk}`, then the embedding should enforce :math:`d_{ij} <
d_{jk}`. A simple algorithm to enforce that is to use a monotonic regression
of :math:`d_{ij}` on :math:`S_{ij}`, yielding disparities :math:`\hat{d}_{ij}`
in the same order as :math:`S_{ij}`.

A trivial solution to this problem is to set all the points on the origin. In
order to avoid that, the disparities :math:`\hat{d}_{ij}` are normalized.


.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_mds_001.png
   :target: ../auto_examples/manifold/plot_mds.html
   :align: center
   :scale: 60


.. topic:: References:

  * `"Modern Multidimensional Scaling - Theory and Applications"
    <https://www.springer.com/fr/book/9780387251509>`_
    Borg, I.; Groenen P. Springer Series in Statistics (1997)

  * `"Nonmetric multidimensional scaling: a numerical method"
    <https://link.springer.com/article/10.1007%2FBF02289694>`_
    Kruskal, J. Psychometrika, 29 (1964)

  * `"Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis"
    <https://link.springer.com/article/10.1007%2FBF02289565>`_
    Kruskal, J. Psychometrika, 29, (1964)

.. _t_sne:

t-distributed Stochastic Neighbor Embedding (t-SNE)
===================================================

t-SNE (:class:`TSNE`) converts affinities of data points to probabilities.
The affinities in the original space are represented by Gaussian joint
probabilities and the affinities in the embedded space are represented by
Student's t-distributions. This allows t-SNE to be particularly sensitive
to local structure and has a few other advantages over existing techniques:

* Revealing the structure at many scales on a single map
* Revealing data that lie in multiple, different, manifolds or clusters
* Reducing the tendency to crowd points together at the center

While Isomap, LLE and variants are best suited to unfold a single continuous
low dimensional manifold, t-SNE will focus on the local structure of the data
and will tend to extract clustered local groups of samples as highlighted on
the S-curve example. This ability to group samples based on the local structure
might be beneficial to visually disentangle a dataset that comprises several
manifolds at once as is the case in the digits dataset.

The Kullback-Leibler (KL) divergence of the joint
probabilities in the original space and the embedded space will be minimized
by gradient descent. Note that the KL divergence is not convex, i.e.
multiple restarts with different initializations will end up in local minima
of the KL divergence. Hence, it is sometimes useful to try different seeds
and select the embedding with the lowest KL divergence.

The disadvantages to using t-SNE are roughly:

* t-SNE is computationally expensive, and can take several hours on million-sample
  datasets where PCA will finish in seconds or minutes
* The Barnes-Hut t-SNE method is limited to two or three dimensional embeddings.
* The algorithm is stochastic and multiple restarts with different seeds can
  yield different embeddings. However, it is perfectly legitimate to pick the
  embedding with the least error.
* Global structure is not explicitly preserved. This problem is mitigated by
  initializing points with PCA (using `init='pca'`).


.. figure:: ../auto_examples/manifold/images/sphx_glr_plot_lle_digits_013.png
   :target: ../auto_examples/manifold/plot_lle_digits.html
   :align: center
   :scale: 50

Optimizing t-SNE
----------------
The main purpose of t-SNE is visualization of high-dimensional data. Hence,
it works best when the data will be embedded on two or three dimensions.

Optimizing the KL divergence can be a little bit tricky sometimes. There are
five parameters that control the optimization of t-SNE and therefore possibly
the quality of the resulting embedding:

* perplexity
* early exaggeration factor
* learning rate
* maximum number of iterations
* angle (not used in the exact method)

The perplexity is defined as :math:`k=2^{(S)}` where :math:`S` is the Shannon
entropy of the conditional probability distribution. The perplexity of a
:math:`k`-sided die is :math:`k`, so that :math:`k` is effectively the number of
nearest neighbors t-SNE considers when generating the conditional probabilities.
Larger perplexities lead to more nearest neighbors and less sensitive to small
structure. Conversely a lower perplexity considers a smaller number of
neighbors, and thus ignores more global information in favour of the
local neighborhood. As dataset sizes get larger more points will be
required to get a reasonable sample of the local neighborhood, and hence
larger perplexities may be required. Similarly noisier datasets will require
larger perplexity values to encompass enough local neighbors to see beyond
the background noise.

The maximum number of iterations is usually high enough and does not need
any tuning. The optimization consists of two phases: the early exaggeration
phase and the final optimization. During early exaggeration the joint
probabilities in the original space will be artificially increased by
multiplication with a given factor. Larger factors result in larger gaps
between natural clusters in the data. If the factor is too high, the KL
divergence could increase during this phase. Usually it does not have to be
tuned. A critical parameter is the learning rate. If it is too low gradient
descent will get stuck in a bad local minimum. If it is too high the KL
divergence will increase during optimization. More tips can be found in
Laurens van der Maaten's FAQ (see references). The last parameter, angle,
is a tradeoff between performance and accuracy. Larger angles imply that we
can approximate larger regions by a single point, leading to better speed
but less accurate results.

`"How to Use t-SNE Effectively" <https://distill.pub/2016/misread-tsne/>`_
provides a good discussion of the effects of the various parameters, as well
as interactive plots to explore the effects of different parameters.

Barnes-Hut t-SNE
----------------

The Barnes-Hut t-SNE that has been implemented here is usually much slower than
other manifold learning algorithms. The optimization is quite difficult
and the computation of the gradient is :math:`O[d N log(N)]`, where :math:`d`
is the number of output dimensions and :math:`N` is the number of samples. The
Barnes-Hut method improves on the exact method where t-SNE complexity is
:math:`O[d N^2]`, but has several other notable differences:

* The Barnes-Hut implementation only works when the target dimensionality is 3
  or less. The 2D case is typical when building visualizations.
* Barnes-Hut only works with dense input data. Sparse data matrices can only be
  embedded with the exact method or can be approximated by a dense low rank
  projection for instance using :class:`sklearn.decomposition.TruncatedSVD`
* Barnes-Hut is an approximation of the exact method. The approximation is
  parameterized with the angle parameter, therefore the angle parameter is
  unused when method="exact"
* Barnes-Hut is significantly more scalable. Barnes-Hut can be used to embed
  hundred of thousands of data points while the exact method can handle
  thousands of samples before becoming computationally intractable

For visualization purpose (which is the main use case of t-SNE), using the
Barnes-Hut method is strongly recommended. The exact t-SNE method is useful
for checking the theoretically properties of the embedding possibly in higher
dimensional space but limit to small datasets due to computational constraints.

Also note that the digits labels roughly match the natural grouping found by
t-SNE while the linear 2D projection of the PCA model yields a representation
where label regions largely overlap. This is a strong clue that this data can
be well separated by non linear methods that focus on the local structure (e.g.
an SVM with a Gaussian RBF kernel). However, failing to visualize well
separated homogeneously labeled groups with t-SNE in 2D does not necessarily
imply that the data cannot be correctly classified by a supervised model. It
might be the case that 2 dimensions are not low enough to accurately represents
the internal structure of the data.


.. topic:: References:

  * `"Visualizing High-Dimensional Data Using t-SNE"
    <http://jmlr.org/papers/v9/vandermaaten08a.html>`_
    van der Maaten, L.J.P.; Hinton, G. Journal of Machine Learning Research
    (2008)

  * `"t-Distributed Stochastic Neighbor Embedding"
    <https://lvdmaaten.github.io/tsne/>`_
    van der Maaten, L.J.P.

  * `"Accelerating t-SNE using Tree-Based Algorithms."
    <https://lvdmaaten.github.io/publications/papers/JMLR_2014.pdf>`_
    L.J.P. van der Maaten.  Journal of Machine Learning Research 15(Oct):3221-3245, 2014.

Tips on practical use
=====================

* Make sure the same scale is used over all features. Because manifold
  learning methods are based on a nearest-neighbor search, the algorithm
  may perform poorly otherwise.  See :ref:`StandardScaler <preprocessing_scaler>`
  for convenient ways of scaling heterogeneous data.

* The reconstruction error computed by each routine can be used to choose
  the optimal output dimension.  For a :math:`d`-dimensional manifold embedded
  in a :math:`D`-dimensional parameter space, the reconstruction error will
  decrease as ``n_components`` is increased until ``n_components == d``.

* Note that noisy data can "short-circuit" the manifold, in essence acting
  as a bridge between parts of the manifold that would otherwise be
  well-separated.  Manifold learning on noisy and/or incomplete data is
  an active area of research.

* Certain input configurations can lead to singular weight matrices, for
  example when more than two points in the dataset are identical, or when
  the data is split into disjointed groups.  In this case, ``solver='arpack'``
  will fail to find the null space.  The easiest way to address this is to
  use ``solver='dense'`` which will work on a singular matrix, though it may
  be very slow depending on the number of input points.  Alternatively, one
  can attempt to understand the source of the singularity: if it is due to
  disjoint sets, increasing ``n_neighbors`` may help.  If it is due to
  identical points in the dataset, removing these points may help.

.. seealso::

   :ref:`random_trees_embedding` can also be useful to derive non-linear
   representations of feature space, also it does not perform
   dimensionality reduction.
