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<li class="toctree-l1 has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
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<section id="manifold-learning">
<span id="manifold"></span><h1><span class="section-number">2.2. </span>Manifold learning<a class="headerlink" href="#manifold-learning" title="Link to this heading">#</a></h1>
<div class="line-block">
<div class="line">Look for the bare necessities</div>
<div class="line">The simple bare necessities</div>
<div class="line">Forget about your worries and your strife</div>
<div class="line">I mean the bare necessities</div>
<div class="line">Old Mother Nature’s recipes</div>
<div class="line">That bring the bare necessities of life</div>
<div class="line"><br /></div>
<div class="line-block">
<div class="line">– Baloo’s song [The Jungle Book]</div>
</div>
</div>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_compare_methods.html"><img alt="../_images/sphx_glr_plot_compare_methods_001.png" src="../_images/sphx_glr_plot_compare_methods_001.png" style="width: 420.0px; height: 420.0px;" />
</a>
</figure>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/manifold/plot_compare_methods.html"><img alt="manifold_img3" src="../_images/sphx_glr_plot_compare_methods_003.png" style="width: 180.0px; height: 180.0px;" /></a> <a class="reference external" href="../auto_examples/manifold/plot_compare_methods.html"><img alt="manifold_img4" src="../_images/sphx_glr_plot_compare_methods_004.png" style="width: 180.0px; height: 180.0px;" /></a> <a class="reference external" href="../auto_examples/manifold/plot_compare_methods.html"><img alt="manifold_img5" src="../_images/sphx_glr_plot_compare_methods_005.png" style="width: 180.0px; height: 180.0px;" /></a> <a class="reference external" href="../auto_examples/manifold/plot_compare_methods.html"><img alt="manifold_img6" src="../_images/sphx_glr_plot_compare_methods_006.png" style="width: 180.0px; height: 180.0px;" /></a></strong></p><p>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.</p>
<section id="introduction">
<h2><span class="section-number">2.2.1. </span>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
<p>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.</p>
<p>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.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="digits_img" src="../_images/sphx_glr_plot_lle_digits_001.png" style="width: 300.0px; height: 300.0px;" /></a> <a class="reference external" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="projected_img" src="../_images/sphx_glr_plot_lle_digits_002.png" style="width: 320.0px; height: 240.0px;" /></a></strong></p><p>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.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="PCA_img" src="../_images/sphx_glr_plot_lle_digits_003.png" style="width: 320.0px; height: 240.0px;" /></a> <a class="reference external" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="LDA_img" src="../_images/sphx_glr_plot_lle_digits_004.png" style="width: 320.0px; height: 240.0px;" /></a></strong></p><p>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.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p>See <a class="reference internal" href="../auto_examples/manifold/plot_lle_digits.html#sphx-glr-auto-examples-manifold-plot-lle-digits-py"><span class="std std-ref">Manifold learning on handwritten digits: Locally Linear Embedding, Isomap…</span></a> for an example of
dimensionality reduction on handwritten digits.</p></li>
<li><p>See <a class="reference internal" href="../auto_examples/manifold/plot_compare_methods.html#sphx-glr-auto-examples-manifold-plot-compare-methods-py"><span class="std std-ref">Comparison of Manifold Learning methods</span></a> for an example of
dimensionality reduction on a toy “S-curve” dataset.</p></li>
<li><p>See <a class="reference internal" href="../auto_examples/applications/plot_stock_market.html#sphx-glr-auto-examples-applications-plot-stock-market-py"><span class="std std-ref">Visualizing the stock market structure</span></a> for an example of
using manifold learning to map the stock market structure based on historical stock
prices.</p></li>
</ul>
<p>The manifold learning implementations available in scikit-learn are
summarized below</p>
</section>
<section id="isomap">
<span id="id1"></span><h2><span class="section-number">2.2.2. </span>Isomap<a class="headerlink" href="#isomap" title="Link to this heading">#</a></h2>
<p>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
<a class="reference internal" href="generated/sklearn.manifold.Isomap.html#sklearn.manifold.Isomap" title="sklearn.manifold.Isomap"><code class="xref py py-class docutils literal notranslate"><span class="pre">Isomap</span></code></a>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="../_images/sphx_glr_plot_lle_digits_005.png" src="../_images/sphx_glr_plot_lle_digits_005.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The Isomap algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Nearest neighbor search.</strong> Isomap uses
<a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a> for efficient neighbor search.
The cost is approximately <span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)]\)</span>, for <span class="math notranslate nohighlight">\(k\)</span>
nearest neighbors of <span class="math notranslate nohighlight">\(N\)</span> points in <span class="math notranslate nohighlight">\(D\)</span> dimensions.</p></li>
<li><p class="sd-card-text"><strong>Shortest-path graph search.</strong> The most efficient known algorithms
for this are <em>Dijkstra’s Algorithm</em>, which is approximately
<span class="math notranslate nohighlight">\(O[N^2(k + \log(N))]\)</span>, or the <em>Floyd-Warshall algorithm</em>, which
is <span class="math notranslate nohighlight">\(O[N^3]\)</span>. The algorithm can be selected by the user with
the <code class="docutils literal notranslate"><span class="pre">path_method</span></code> keyword of <code class="docutils literal notranslate"><span class="pre">Isomap</span></code>. If unspecified, the code
attempts to choose the best algorithm for the input data.</p></li>
<li><p class="sd-card-text"><strong>Partial eigenvalue decomposition.</strong> The embedding is encoded in the
eigenvectors corresponding to the <span class="math notranslate nohighlight">\(d\)</span> largest eigenvalues of the
<span class="math notranslate nohighlight">\(N \times N\)</span> isomap kernel. For a dense solver, the cost is
approximately <span class="math notranslate nohighlight">\(O[d N^2]\)</span>. This cost can often be improved using
the <code class="docutils literal notranslate"><span class="pre">ARPACK</span></code> solver. The eigensolver can be specified by the user
with the <code class="docutils literal notranslate"><span class="pre">eigen_solver</span></code> keyword of <code class="docutils literal notranslate"><span class="pre">Isomap</span></code>. If unspecified, the
code attempts to choose the best algorithm for the input data.</p></li>
</ol>
<p class="sd-card-text">The overall complexity of Isomap is
<span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)] + O[N^2(k + \log(N))] + O[d N^2]\)</span>.</p>
<ul class="simple">
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(N\)</span> : number of training data points</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(D\)</span> : input dimension</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(k\)</span> : number of nearest neighbors</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(d\)</span> : output dimension</p></li>
</ul>
</div>
</details><p class="rubric">References</p>
<ul class="simple">
<li><p><a class="reference external" href="http://science.sciencemag.org/content/290/5500/2319.full">“A global geometric framework for nonlinear dimensionality reduction”</a>
Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. Science 290 (5500)</p></li>
</ul>
</section>
<section id="locally-linear-embedding">
<span id="id2"></span><h2><span class="section-number">2.2.3. </span>Locally Linear Embedding<a class="headerlink" href="#locally-linear-embedding" title="Link to this heading">#</a></h2>
<p>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.</p>
<p>Locally linear embedding can be performed with function
<a class="reference internal" href="generated/sklearn.manifold.locally_linear_embedding.html#sklearn.manifold.locally_linear_embedding" title="sklearn.manifold.locally_linear_embedding"><code class="xref py py-func docutils literal notranslate"><span class="pre">locally_linear_embedding</span></code></a> or its object-oriented counterpart
<a class="reference internal" href="generated/sklearn.manifold.LocallyLinearEmbedding.html#sklearn.manifold.LocallyLinearEmbedding" title="sklearn.manifold.LocallyLinearEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">LocallyLinearEmbedding</span></code></a>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="../_images/sphx_glr_plot_lle_digits_006.png" src="../_images/sphx_glr_plot_lle_digits_006.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity-2">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity-2" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The standard LLE algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Nearest Neighbors Search</strong>. See discussion under Isomap above.</p></li>
<li><p class="sd-card-text"><strong>Weight Matrix Construction</strong>. <span class="math notranslate nohighlight">\(O[D N k^3]\)</span>.
The construction of the LLE weight matrix involves the solution of a
<span class="math notranslate nohighlight">\(k \times k\)</span> linear equation for each of the <span class="math notranslate nohighlight">\(N\)</span> local
neighborhoods.</p></li>
<li><p class="sd-card-text"><strong>Partial Eigenvalue Decomposition</strong>. See discussion under Isomap above.</p></li>
</ol>
<p class="sd-card-text">The overall complexity of standard LLE is
<span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)] + O[D N k^3] + O[d N^2]\)</span>.</p>
<ul class="simple">
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(N\)</span> : number of training data points</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(D\)</span> : input dimension</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(k\)</span> : number of nearest neighbors</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(d\)</span> : output dimension</p></li>
</ul>
</div>
</details><p class="rubric">References</p>
<ul class="simple">
<li><p><a class="reference external" href="http://www.sciencemag.org/content/290/5500/2323.full">“Nonlinear dimensionality reduction by locally linear embedding”</a>
Roweis, S. & Saul, L. Science 290:2323 (2000)</p></li>
</ul>
</section>
<section id="modified-locally-linear-embedding">
<h2><span class="section-number">2.2.4. </span>Modified Locally Linear Embedding<a class="headerlink" href="#modified-locally-linear-embedding" title="Link to this heading">#</a></h2>
<p>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 <span class="math notranslate nohighlight">\(r\)</span>, which is chosen
relative to the trace of the local weight matrix. Though it can be shown
formally that as <span class="math notranslate nohighlight">\(r \to 0\)</span>, the solution converges to the desired
embedding, there is no guarantee that the optimal solution will be found
for <span class="math notranslate nohighlight">\(r > 0\)</span>. This problem manifests itself in embeddings which distort
the underlying geometry of the manifold.</p>
<p>One method to address the regularization problem is to use multiple weight
vectors in each neighborhood. This is the essence of <em>modified locally
linear embedding</em> (MLLE). MLLE can be performed with function
<a class="reference internal" href="generated/sklearn.manifold.locally_linear_embedding.html#sklearn.manifold.locally_linear_embedding" title="sklearn.manifold.locally_linear_embedding"><code class="xref py py-func docutils literal notranslate"><span class="pre">locally_linear_embedding</span></code></a> or its object-oriented counterpart
<a class="reference internal" href="generated/sklearn.manifold.LocallyLinearEmbedding.html#sklearn.manifold.LocallyLinearEmbedding" title="sklearn.manifold.LocallyLinearEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">LocallyLinearEmbedding</span></code></a>, with the keyword <code class="docutils literal notranslate"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'modified'</span></code>.
It requires <code class="docutils literal notranslate"><span class="pre">n_neighbors</span> <span class="pre">></span> <span class="pre">n_components</span></code>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="../_images/sphx_glr_plot_lle_digits_007.png" src="../_images/sphx_glr_plot_lle_digits_007.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity-3">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity-3" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The MLLE algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Nearest Neighbors Search</strong>. Same as standard LLE</p></li>
<li><p class="sd-card-text"><strong>Weight Matrix Construction</strong>. Approximately
<span class="math notranslate nohighlight">\(O[D N k^3] + O[N (k-D) k^2]\)</span>. 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.</p></li>
<li><p class="sd-card-text"><strong>Partial Eigenvalue Decomposition</strong>. Same as standard LLE</p></li>
</ol>
<p class="sd-card-text">The overall complexity of MLLE is
<span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)] + O[D N k^3] + O[N (k-D) k^2] + O[d N^2]\)</span>.</p>
<ul class="simple">
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(N\)</span> : number of training data points</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(D\)</span> : input dimension</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(k\)</span> : number of nearest neighbors</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(d\)</span> : output dimension</p></li>
</ul>
</div>
</details><p class="rubric">References</p>
<ul class="simple">
<li><p><a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/0b060fdbd92cbcc66b383bcaa9ba5e5e624d7ee3">“MLLE: Modified Locally Linear Embedding Using Multiple Weights”</a>
Zhang, Z. & Wang, J.</p></li>
</ul>
</section>
<section id="hessian-eigenmapping">
<h2><span class="section-number">2.2.5. </span>Hessian Eigenmapping<a class="headerlink" href="#hessian-eigenmapping" title="Link to this heading">#</a></h2>
<p>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, <code class="docutils literal notranslate"><span class="pre">sklearn</span></code> 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
<a class="reference internal" href="generated/sklearn.manifold.locally_linear_embedding.html#sklearn.manifold.locally_linear_embedding" title="sklearn.manifold.locally_linear_embedding"><code class="xref py py-func docutils literal notranslate"><span class="pre">locally_linear_embedding</span></code></a> or its object-oriented counterpart
<a class="reference internal" href="generated/sklearn.manifold.LocallyLinearEmbedding.html#sklearn.manifold.LocallyLinearEmbedding" title="sklearn.manifold.LocallyLinearEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">LocallyLinearEmbedding</span></code></a>, with the keyword <code class="docutils literal notranslate"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'hessian'</span></code>.
It requires <code class="docutils literal notranslate"><span class="pre">n_neighbors</span> <span class="pre">></span> <span class="pre">n_components</span> <span class="pre">*</span> <span class="pre">(n_components</span> <span class="pre">+</span> <span class="pre">3)</span> <span class="pre">/</span> <span class="pre">2</span></code>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="../_images/sphx_glr_plot_lle_digits_008.png" src="../_images/sphx_glr_plot_lle_digits_008.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity-4">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity-4" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The HLLE algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Nearest Neighbors Search</strong>. Same as standard LLE</p></li>
<li><p class="sd-card-text"><strong>Weight Matrix Construction</strong>. Approximately
<span class="math notranslate nohighlight">\(O[D N k^3] + O[N d^6]\)</span>. 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.</p></li>
<li><p class="sd-card-text"><strong>Partial Eigenvalue Decomposition</strong>. Same as standard LLE.</p></li>
</ol>
<p class="sd-card-text">The overall complexity of standard HLLE is
<span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)] + O[D N k^3] + O[N d^6] + O[d N^2]\)</span>.</p>
<ul class="simple">
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(N\)</span> : number of training data points</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(D\)</span> : input dimension</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(k\)</span> : number of nearest neighbors</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(d\)</span> : output dimension</p></li>
</ul>
</div>
</details><p class="rubric">References</p>
<ul class="simple">
<li><p><a class="reference external" href="http://www.pnas.org/content/100/10/5591">“Hessian Eigenmaps: Locally linear embedding techniques for
high-dimensional data”</a>
Donoho, D. & Grimes, C. Proc Natl Acad Sci USA. 100:5591 (2003)</p></li>
</ul>
</section>
<section id="spectral-embedding">
<span id="id3"></span><h2><span class="section-number">2.2.6. </span>Spectral Embedding<a class="headerlink" href="#spectral-embedding" title="Link to this heading">#</a></h2>
<p>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 <a class="reference internal" href="generated/sklearn.manifold.spectral_embedding.html#sklearn.manifold.spectral_embedding" title="sklearn.manifold.spectral_embedding"><code class="xref py py-func docutils literal notranslate"><span class="pre">spectral_embedding</span></code></a> or its object-oriented counterpart
<a class="reference internal" href="generated/sklearn.manifold.SpectralEmbedding.html#sklearn.manifold.SpectralEmbedding" title="sklearn.manifold.SpectralEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralEmbedding</span></code></a>.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity-5">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity-5" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The Spectral Embedding (Laplacian Eigenmaps) algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Weighted Graph Construction</strong>. Transform the raw input data into
graph representation using affinity (adjacency) matrix representation.</p></li>
<li><p class="sd-card-text"><strong>Graph Laplacian Construction</strong>. unnormalized Graph Laplacian
is constructed as <span class="math notranslate nohighlight">\(L = D - A\)</span> for and normalized one as
<span class="math notranslate nohighlight">\(L = D^{-\frac{1}{2}} (D - A) D^{-\frac{1}{2}}\)</span>.</p></li>
<li><p class="sd-card-text"><strong>Partial Eigenvalue Decomposition</strong>. Eigenvalue decomposition is
done on graph Laplacian.</p></li>
</ol>
<p class="sd-card-text">The overall complexity of spectral embedding is
<span class="math notranslate nohighlight">\(O[D \log(k) N \log(N)] + O[D N k^3] + O[d N^2]\)</span>.</p>
<ul class="simple">
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(N\)</span> : number of training data points</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(D\)</span> : input dimension</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(k\)</span> : number of nearest neighbors</p></li>
<li><p class="sd-card-text"><span class="math notranslate nohighlight">\(d\)</span> : output dimension</p></li>
</ul>
</div>
</details><p class="rubric">References</p>
<ul class="simple">
<li><p><a class="reference external" href="https://web.cse.ohio-state.edu/~mbelkin/papers/LEM_NC_03.pdf">“Laplacian Eigenmaps for Dimensionality Reduction
and Data Representation”</a>
M. Belkin, P. Niyogi, Neural Computation, June 2003; 15 (6):1373-1396</p></li>
</ul>
</section>
<section id="local-tangent-space-alignment">
<h2><span class="section-number">2.2.7. </span>Local Tangent Space Alignment<a class="headerlink" href="#local-tangent-space-alignment" title="Link to this heading">#</a></h2>
<p>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
<a class="reference internal" href="generated/sklearn.manifold.locally_linear_embedding.html#sklearn.manifold.locally_linear_embedding" title="sklearn.manifold.locally_linear_embedding"><code class="xref py py-func docutils literal notranslate"><span class="pre">locally_linear_embedding</span></code></a> or its object-oriented counterpart
<a class="reference internal" href="generated/sklearn.manifold.LocallyLinearEmbedding.html#sklearn.manifold.LocallyLinearEmbedding" title="sklearn.manifold.LocallyLinearEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">LocallyLinearEmbedding</span></code></a>, with the keyword <code class="docutils literal notranslate"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'ltsa'</span></code>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/manifold/plot_lle_digits.html"><img alt="../_images/sphx_glr_plot_lle_digits_009.png" src="../_images/sphx_glr_plot_lle_digits_009.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="complexity-6">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Complexity<a class="headerlink" href="#complexity-6" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The LTSA algorithm comprises three stages:</p>
<ol class="arabic simple">
<li><p class="sd-card-text"><strong>Nearest Neighbors Search</strong>. Same as standard LLE</p></li>
<li><p class="sd-card-text"><strong>Weight Matrix Construction</strong>. Approximately
<span class="math notranslate nohighlight">\(O[D N k^3] + O[k^2 d]\)</span>. The first term reflects a similar
cost to that of standard LLE.</p></li>
<li><p class="sd-card-text"><strong>Partial Eigenvalue Decomposition</strong>. Same as standard LLE</p></li>
</ol>
<p class="sd-card-text">The overall complexity of standard LTSA is