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<li><a class="reference internal" href="#">The Johnson-Lindenstrauss bound for embedding with random projections</a><ul>
<li><a class="reference internal" href="#theoretical-bounds">Theoretical bounds</a></li>
<li><a class="reference internal" href="#empirical-validation">Empirical validation</a></li>
<li><a class="reference internal" href="#remarks">Remarks</a></li>
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<div class="sphx-glr-example-title section" id="the-johnson-lindenstrauss-bound-for-embedding-with-random-projections">
<span id="sphx-glr-auto-examples-plot-johnson-lindenstrauss-bound-py"></span><h1>The Johnson-Lindenstrauss bound for embedding with random projections<a class="headerlink" href="#the-johnson-lindenstrauss-bound-for-embedding-with-random-projections" title="Permalink to this headline">¶</a></h1>
<p>The <a class="reference external" href="https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma">Johnson-Lindenstrauss lemma</a> states that any high dimensional
dataset can be randomly projected into a lower dimensional Euclidean
space while controlling the distortion in the pairwise distances.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">sys</span>
<span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="View documentation for time.time"><span class="n">time</span></a>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">distutils.version</span> <span class="kn">import</span> <span class="n">LooseVersion</span>
<span class="kn">from</span> <span class="nn">sklearn.random_projection</span> <span class="kn">import</span> <a href="../modules/generated/sklearn.random_projection.johnson_lindenstrauss_min_dim.html#sklearn.random_projection.johnson_lindenstrauss_min_dim" title="View documentation for sklearn.random_projection.johnson_lindenstrauss_min_dim"><span class="n">johnson_lindenstrauss_min_dim</span></a>
<span class="kn">from</span> <span class="nn">sklearn.random_projection</span> <span class="kn">import</span> <a href="../modules/generated/sklearn.random_projection.SparseRandomProjection.html#sklearn.random_projection.SparseRandomProjection" title="View documentation for sklearn.random_projection.SparseRandomProjection"><span class="n">SparseRandomProjection</span></a>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../modules/generated/sklearn.datasets.fetch_20newsgroups_vectorized.html#sklearn.datasets.fetch_20newsgroups_vectorized" title="View documentation for sklearn.datasets.fetch_20newsgroups_vectorized"><span class="n">fetch_20newsgroups_vectorized</span></a>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../modules/generated/sklearn.datasets.load_digits.html#sklearn.datasets.load_digits" title="View documentation for sklearn.datasets.load_digits"><span class="n">load_digits</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics.pairwise</span> <span class="kn">import</span> <span class="n">euclidean_distances</span>
<span class="c1"># `normed` is being deprecated in favor of `density` in histograms</span>
<span class="k">if</span> <span class="n">LooseVersion</span><span class="p">(</span><span class="n">matplotlib</span><span class="o">.</span><span class="n">__version__</span><span class="p">)</span> <span class="o">>=</span> <span class="s1">'2.1'</span><span class="p">:</span>
<span class="n">density_param</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'density'</span><span class="p">:</span> <span class="kc">True</span><span class="p">}</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">density_param</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'normed'</span><span class="p">:</span> <span class="kc">True</span><span class="p">}</span>
</pre></div>
</div>
<div class="section" id="theoretical-bounds">
<h2>Theoretical bounds<a class="headerlink" href="#theoretical-bounds" title="Permalink to this headline">¶</a></h2>
<p>The distortion introduced by a random projection <code class="docutils literal notranslate"><span class="pre">p</span></code> is asserted by
the fact that <code class="docutils literal notranslate"><span class="pre">p</span></code> is defining an eps-embedding with good probability
as defined by:</p>
<div class="math notranslate nohighlight">
\[(1 - eps) \|u - v\|^2 < \|p(u) - p(v)\|^2 < (1 + eps) \|u - v\|^2\]</div>
<p>Where u and v are any rows taken from a dataset of shape [n_samples,
n_features] and p is a projection by a random Gaussian N(0, 1) matrix
with shape [n_components, n_features] (or a sparse Achlioptas matrix).</p>
<p>The minimum number of components to guarantees the eps-embedding is
given by:</p>
<div class="math notranslate nohighlight">
\[n\_components >= 4 log(n\_samples) / (eps^2 / 2 - eps^3 / 3)\]</div>
<p>The first plot shows that with an increasing number of samples <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>,
the minimal number of dimensions <code class="docutils literal notranslate"><span class="pre">n_components</span></code> increased logarithmically
in order to guarantee an <code class="docutils literal notranslate"><span class="pre">eps</span></code>-embedding.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># range of admissible distortions</span>
<span class="n">eps_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Blues</span><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">eps_range</span><span class="p">)))</span>
<span class="c1"># range of number of samples (observation) to embed</span>
<span class="n">n_samples_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.logspace.html#numpy.logspace" title="View documentation for numpy.logspace"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<span class="k">for</span> <span class="n">eps</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">eps_range</span><span class="p">,</span> <span class="n">colors</span><span class="p">):</span>
<span class="n">min_n_components</span> <span class="o">=</span> <a href="../modules/generated/sklearn.random_projection.johnson_lindenstrauss_min_dim.html#sklearn.random_projection.johnson_lindenstrauss_min_dim" title="View documentation for sklearn.random_projection.johnson_lindenstrauss_min_dim"><span class="n">johnson_lindenstrauss_min_dim</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.loglog.html#matplotlib.pyplot.loglog" title="View documentation for matplotlib.pyplot.loglog"><span class="n">plt</span><span class="o">.</span><span class="n">loglog</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">min_n_components</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">([</span><span class="s2">"eps = </span><span class="si">%0.1f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">eps</span> <span class="k">for</span> <span class="n">eps</span> <span class="ow">in</span> <span class="n">eps_range</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Number of observations to eps-embed"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Minimum number of dimensions"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Johnson-Lindenstrauss bounds:</span><span class="se">\n</span><span class="s2">n_samples vs n_components"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img alt="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_001.png" class="sphx-glr-single-img" src="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_001.png" />
<p>The second plot shows that an increase of the admissible
distortion <code class="docutils literal notranslate"><span class="pre">eps</span></code> allows to reduce drastically the minimal number of
dimensions <code class="docutils literal notranslate"><span class="pre">n_components</span></code> for a given number of samples <code class="docutils literal notranslate"><span class="pre">n_samples</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># range of admissible distortions</span>
<span class="n">eps_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="c1"># range of number of samples (observation) to embed</span>
<span class="n">n_samples_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.logspace.html#numpy.logspace" title="View documentation for numpy.logspace"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Blues</span><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html#numpy.linspace" title="View documentation for numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">n_samples_range</span><span class="p">)))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<span class="k">for</span> <span class="n">n_samples</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">colors</span><span class="p">):</span>
<span class="n">min_n_components</span> <span class="o">=</span> <a href="../modules/generated/sklearn.random_projection.johnson_lindenstrauss_min_dim.html#sklearn.random_projection.johnson_lindenstrauss_min_dim" title="View documentation for sklearn.random_projection.johnson_lindenstrauss_min_dim"><span class="n">johnson_lindenstrauss_min_dim</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps_range</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.semilogy.html#matplotlib.pyplot.semilogy" title="View documentation for matplotlib.pyplot.semilogy"><span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span></a><span class="p">(</span><span class="n">eps_range</span><span class="p">,</span> <span class="n">min_n_components</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">([</span><span class="s2">"n_samples = </span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span> <span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">n_samples_range</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper right"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Distortion eps"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Minimum number of dimensions"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Johnson-Lindenstrauss bounds:</span><span class="se">\n</span><span class="s2">n_components vs eps"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img alt="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_002.png" class="sphx-glr-single-img" src="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_002.png" />
</div>
<div class="section" id="empirical-validation">
<h2>Empirical validation<a class="headerlink" href="#empirical-validation" title="Permalink to this headline">¶</a></h2>
<p>We validate the above bounds on the 20 newsgroups text document
(TF-IDF word frequencies) dataset or on the digits dataset:</p>
<ul class="simple">
<li><p>for the 20 newsgroups dataset some 500 documents with 100k
features in total are projected using a sparse random matrix to smaller
euclidean spaces with various values for the target number of dimensions
<code class="docutils literal notranslate"><span class="pre">n_components</span></code>.</p></li>
<li><p>for the digits dataset, some 8x8 gray level pixels data for 500
handwritten digits pictures are randomly projected to spaces for various
larger number of dimensions <code class="docutils literal notranslate"><span class="pre">n_components</span></code>.</p></li>
</ul>
<p>The default dataset is the 20 newsgroups dataset. To run the example on the
digits dataset, pass the <code class="docutils literal notranslate"><span class="pre">--use-digits-dataset</span></code> command line argument to
this script.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">if</span> <span class="s1">'--use-digits-dataset'</span> <span class="ow">in</span> <a href="https://docs.python.org/3/library/sys.html#sys.argv" title="View documentation for sys.argv"><span class="n">sys</span><span class="o">.</span><span class="n">argv</span></a><span class="p">:</span>
<span class="n">data</span> <span class="o">=</span> <a href="../modules/generated/sklearn.datasets.load_digits.html#sklearn.datasets.load_digits" title="View documentation for sklearn.datasets.load_digits"><span class="n">load_digits</span></a><span class="p">()</span><span class="o">.</span><span class="n">data</span><span class="p">[:</span><span class="mi">500</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">data</span> <span class="o">=</span> <a href="../modules/generated/sklearn.datasets.fetch_20newsgroups_vectorized.html#sklearn.datasets.fetch_20newsgroups_vectorized" title="View documentation for sklearn.datasets.fetch_20newsgroups_vectorized"><span class="n">fetch_20newsgroups_vectorized</span></a><span class="p">()</span><span class="o">.</span><span class="n">data</span><span class="p">[:</span><span class="mi">500</span><span class="p">]</span>
</pre></div>
</div>
<p>For each value of <code class="docutils literal notranslate"><span class="pre">n_components</span></code>, we plot:</p>
<ul class="simple">
<li><p>2D distribution of sample pairs with pairwise distances in original
and projected spaces as x and y axis respectively.</p></li>
<li><p>1D histogram of the ratio of those distances (projected / original).</p></li>
</ul>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Embedding </span><span class="si">%d</span><span class="s2"> samples with dim </span><span class="si">%d</span><span class="s2"> using various random projections"</span>
<span class="o">%</span> <span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="n">n_components_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html#numpy.array" title="View documentation for numpy.array"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([</span><span class="mi">300</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">10000</span><span class="p">])</span>
<span class="n">dists</span> <span class="o">=</span> <span class="n">euclidean_distances</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">squared</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="c1"># select only non-identical samples pairs</span>
<span class="n">nonzero</span> <span class="o">=</span> <span class="n">dists</span> <span class="o">!=</span> <span class="mi">0</span>
<span class="n">dists</span> <span class="o">=</span> <span class="n">dists</span><span class="p">[</span><span class="n">nonzero</span><span class="p">]</span>
<span class="k">for</span> <span class="n">n_components</span> <span class="ow">in</span> <span class="n">n_components_range</span><span class="p">:</span>
<span class="n">t0</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="View documentation for time.time"><span class="n">time</span></a><span class="p">()</span>
<span class="n">rp</span> <span class="o">=</span> <a href="../modules/generated/sklearn.random_projection.SparseRandomProjection.html#sklearn.random_projection.SparseRandomProjection" title="View documentation for sklearn.random_projection.SparseRandomProjection"><span class="n">SparseRandomProjection</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="n">n_components</span><span class="p">)</span>
<span class="n">projected_data</span> <span class="o">=</span> <span class="n">rp</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Projected </span><span class="si">%d</span><span class="s2"> samples from </span><span class="si">%d</span><span class="s2"> to </span><span class="si">%d</span><span class="s2"> in </span><span class="si">%0.3f</span><span class="s2">s"</span>
<span class="o">%</span> <span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">,</span> <span class="n">n_components</span><span class="p">,</span> <a href="https://docs.python.org/3/library/time.html#time.time" title="View documentation for time.time"><span class="n">time</span></a><span class="p">()</span> <span class="o">-</span> <span class="n">t0</span><span class="p">))</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">rp</span><span class="p">,</span> <span class="s1">'components_'</span><span class="p">):</span>
<span class="n">n_bytes</span> <span class="o">=</span> <span class="n">rp</span><span class="o">.</span><span class="n">components_</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">nbytes</span>
<span class="n">n_bytes</span> <span class="o">+=</span> <span class="n">rp</span><span class="o">.</span><span class="n">components_</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">nbytes</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Random matrix with size: </span><span class="si">%0.3f</span><span class="s2">MB"</span> <span class="o">%</span> <span class="p">(</span><span class="n">n_bytes</span> <span class="o">/</span> <span class="mf">1e6</span><span class="p">))</span>
<span class="n">projected_dists</span> <span class="o">=</span> <span class="n">euclidean_distances</span><span class="p">(</span>
<span class="n">projected_data</span><span class="p">,</span> <span class="n">squared</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()[</span><span class="n">nonzero</span><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<span class="n">min_dist</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">projected_dists</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">dists</span><span class="o">.</span><span class="n">min</span><span class="p">())</span>
<span class="n">max_dist</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">projected_dists</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="n">dists</span><span class="o">.</span><span class="n">max</span><span class="p">())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.hexbin.html#matplotlib.pyplot.hexbin" title="View documentation for matplotlib.pyplot.hexbin"><span class="n">plt</span><span class="o">.</span><span class="n">hexbin</span></a><span class="p">(</span><span class="n">dists</span><span class="p">,</span> <span class="n">projected_dists</span><span class="p">,</span> <span class="n">gridsize</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">PuBu</span><span class="p">,</span>
<span class="n">extent</span><span class="o">=</span><span class="p">[</span><span class="n">min_dist</span><span class="p">,</span> <span class="n">max_dist</span><span class="p">,</span> <span class="n">min_dist</span><span class="p">,</span> <span class="n">max_dist</span><span class="p">])</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Pairwise squared distances in original space"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Pairwise squared distances in projected space"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Pairwise distances distribution for n_components=</span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span>
<span class="n">n_components</span><span class="p">)</span>
<span class="n">cb</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.colorbar.html#matplotlib.pyplot.colorbar" title="View documentation for matplotlib.pyplot.colorbar"><span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span></a><span class="p">()</span>
<span class="n">cb</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="s1">'Sample pairs counts'</span><span class="p">)</span>
<span class="n">rates</span> <span class="o">=</span> <span class="n">projected_dists</span> <span class="o">/</span> <span class="n">dists</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Mean distances rate: </span><span class="si">%0.2f</span><span class="s2"> (</span><span class="si">%0.2f</span><span class="s2">)"</span>
<span class="o">%</span> <span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html#numpy.mean" title="View documentation for numpy.mean"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">rates</span><span class="p">),</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html#numpy.std" title="View documentation for numpy.std"><span class="n">np</span><span class="o">.</span><span class="n">std</span></a><span class="p">(</span><span class="n">rates</span><span class="p">)))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.hist.html#matplotlib.pyplot.hist" title="View documentation for matplotlib.pyplot.hist"><span class="n">plt</span><span class="o">.</span><span class="n">hist</span></a><span class="p">(</span><span class="n">rates</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="nb">range</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">),</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">'k'</span><span class="p">,</span> <span class="o">**</span><span class="n">density_param</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Squared distances rate: projected / original"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Distribution of samples pairs"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Histogram of pairwise distance rates for n_components=</span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span>
<span class="n">n_components</span><span class="p">)</span>
<span class="c1"># TODO: compute the expected value of eps and add them to the previous plot</span>
<span class="c1"># as vertical lines / region</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
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<li><img alt="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_003.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_003.png" />
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<li><img alt="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_004.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_004.png" />
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<li><img alt="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_008.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_plot_johnson_lindenstrauss_bound_008.png" />
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</ul>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Embedding 500 samples with dim 130107 using various random projections
Projected 500 samples from 130107 to 300 in 0.632s
Random matrix with size: 1.296MB
Mean distances rate: 1.00 (0.19)
Projected 500 samples from 130107 to 1000 in 2.118s
Random matrix with size: 4.328MB
Mean distances rate: 1.00 (0.10)
Projected 500 samples from 130107 to 10000 in 22.211s
Random matrix with size: 43.315MB
Mean distances rate: 1.03 (0.04)
</pre></div>
</div>
<p>We can see that for low values of <code class="docutils literal notranslate"><span class="pre">n_components</span></code> the distribution is wide
with many distorted pairs and a skewed distribution (due to the hard
limit of zero ratio on the left as distances are always positives)
while for larger values of n_components the distortion is controlled
and the distances are well preserved by the random projection.</p>
</div>
<div class="section" id="remarks">
<h2>Remarks<a class="headerlink" href="#remarks" title="Permalink to this headline">¶</a></h2>
<p>According to the JL lemma, projecting 500 samples without too much distortion
will require at least several thousands dimensions, irrespective of the
number of features of the original dataset.</p>
<p>Hence using random projections on the digits dataset which only has 64
features in the input space does not make sense: it does not allow
for dimensionality reduction in this case.</p>
<p>On the twenty newsgroups on the other hand the dimensionality can be
decreased from 56436 down to 10000 while reasonably preserving
pairwise distances.</p>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 28.844 seconds)</p>
<p><strong>Estimated memory usage:</strong> 96 MB</p>
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