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<section class="sphx-glr-example-title" id="row-and-column-alignments-with-co-optimal-transport">
<span id="sphx-glr-auto-examples-others-plot-coot-py"></span><h1>Row and column alignments with CO-Optimal Transport<a class="headerlink" href="#row-and-column-alignments-with-co-optimal-transport" title="Link to this heading"></a></h1>
<p>This example is designed to show how to use the CO-Optimal Transport <a href="#id3"><span class="problematic" id="id1">[47]_</span></a> in POT.
CO-Optimal Transport allows to calculate the distance between two <strong>arbitrary-size</strong>
matrices, and to align their rows and columns. In this example, we consider two
random matrices <span class="math notranslate nohighlight">\(X_1\)</span> and <span class="math notranslate nohighlight">\(X_2\)</span> defined by
<span class="math notranslate nohighlight">\((X_1)_{i,j} = \cos(\frac{i}{n_1} \pi) + \cos(\frac{j}{d_1} \pi) + \sigma \mathcal N(0,1)\)</span>
and <span class="math notranslate nohighlight">\((X_2)_{i,j} = \cos(\frac{i}{n_2} \pi) + \cos(\frac{j}{d_2} \pi) + \sigma \mathcal N(0,1)\)</span>.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>49<span class="fn-bracket">]</span></span>
<p>Redko, I., Vayer, T., Flamary, R., and Courty, N. (2020).
<a class="reference external" href="https://proceedings.neurips.cc/paper/2020/file/cc384c68ad503482fb24e6d1e3b512ae-Paper.pdf">CO-Optimal Transport</a>.
Advances in Neural Information Processing Systems, 33.</p>
</aside>
</aside>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Remi Flamary <remi.flamary@unice.fr></span>
<span class="c1"># Quang Huy Tran <quang-huy.tran@univ-ubs.fr></span>
<span class="c1"># License: MIT License</span>
<span class="c1"># sphinx_gallery_thumbnail_number = 2</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">matplotlib.patches</span><span class="w"> </span><span class="kn">import</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class"><span class="n">ConnectionPatch</span></a>
<span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pylab</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">pl</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">ot.coot</span><span class="w"> </span><span class="kn">import</span> <span class="n">co_optimal_transport</span> <span class="k">as</span> <a href="../../gen_modules/ot.coot.html#id0" title="ot.coot.co_optimal_transport" class="sphx-glr-backref-module-ot-coot sphx-glr-backref-type-py-function"><span class="n">coot</span></a>
<span class="kn">from</span><span class="w"> </span><span class="nn">ot.coot</span><span class="w"> </span><span class="kn">import</span> <span class="n">co_optimal_transport2</span> <span class="k">as</span> <a href="../../gen_modules/ot.coot.html#id4" title="ot.coot.co_optimal_transport2" class="sphx-glr-backref-module-ot-coot sphx-glr-backref-type-py-function"><span class="n">coot2</span></a>
</pre></div>
</div>
<p>Generating two random matrices</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n1</span></a> <span class="o">=</span> <span class="mi">20</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n2</span></a> <span class="o">=</span> <span class="mi">10</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a> <span class="o">=</span> <span class="mi">16</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d2</span></a> <span class="o">=</span> <span class="mi">8</span>
<a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">sigma</span></a> <span class="o">=</span> <span class="mf">0.2</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X1</span></a> <span class="o">=</span> <span class="p">(</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.ufunc.html#numpy.ufunc" title="numpy.ufunc" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">cos</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n1</span></a><span class="p">)</span> <span class="o">*</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">/</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n1</span></a><span class="p">)[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ufunc.html#numpy.ufunc" title="numpy.ufunc" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">cos</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a><span class="p">)</span> <span class="o">*</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">/</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a><span class="p">)[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span>
<span class="o">+</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">sigma</span></a> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n1</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a><span class="p">)</span>
<span class="p">)</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X2</span></a> <span class="o">=</span> <span class="p">(</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.ufunc.html#numpy.ufunc" title="numpy.ufunc" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">cos</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n2</span></a><span class="p">)</span> <span class="o">*</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">/</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n2</span></a><span class="p">)[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ufunc.html#numpy.ufunc" title="numpy.ufunc" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">cos</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d2</span></a><span class="p">)</span> <span class="o">*</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">/</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d2</span></a><span class="p">)[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span>
<span class="o">+</span> <a href="https://docs.python.org/3/library/functions.html#float" title="builtins.float" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">sigma</span></a> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n2</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d2</span></a><span class="p">)</span>
<span class="p">)</span>
</pre></div>
</div>
<p>Visualizing the matrices</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X1</span></a><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"$X_1$"</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X2</span></a><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"$X_2$"</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_COOT_001.png" srcset="../../_images/sphx_glr_plot_COOT_001.png" alt="$X_1$, $X_2$" class = "sphx-glr-single-img"/><p>Visualizing the alignments of rows and columns, and calculating the CO-Optimal Transport distance</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pi_sample</span></a><span class="p">,</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pi_feature</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/stdtypes.html#dict" title="builtins.dict" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">log</span></a> <span class="o">=</span> <a href="../../gen_modules/ot.coot.html#id0" title="ot.coot.co_optimal_transport" class="sphx-glr-backref-module-ot-coot sphx-glr-backref-type-py-function"><span class="n">coot</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X1</span></a><span class="p">,</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X2</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/stdtypes.html#dict" title="builtins.dict" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">log</span></a><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.float64" title="numpy.float64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">coot_distance</span></a> <span class="o">=</span> <a href="../../gen_modules/ot.coot.html#id4" title="ot.coot.co_optimal_transport2" class="sphx-glr-backref-module-ot-coot sphx-glr-backref-type-py-function"><span class="n">coot2</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X1</span></a><span class="p">,</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X2</span></a><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"CO-Optimal Transport distance = </span><span class="si">{:.5f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.float64" title="numpy.float64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">coot_distance</span></a><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.figure.Figure.html#matplotlib.figure.Figure" title="matplotlib.figure.Figure" class="sphx-glr-backref-module-matplotlib-figure sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">fig</span></a> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">clf</span><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.html#matplotlib.axes.Axes" title="matplotlib.axes.Axes" class="sphx-glr-backref-module-matplotlib-axes sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax1</span></a> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X1</span></a><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$X_1$"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.html#matplotlib.axes.Axes" title="matplotlib.axes.Axes" class="sphx-glr-backref-module-matplotlib-axes sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax2</span></a> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.axis.YAxis.tick_right.html#matplotlib.axis.YAxis.tick_right" title="matplotlib.axis.YAxis.tick_right" class="sphx-glr-backref-module-matplotlib-axis sphx-glr-backref-type-py-method"><span class="n">ax2</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">tick_right</span></a><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.transpose.html#numpy.transpose" title="numpy.transpose" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">transpose</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">X2</span></a><span class="p">))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Transpose($X_2$)"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.axis.XAxis.tick_top.html#matplotlib.axis.XAxis.tick_top" title="matplotlib.axis.XAxis.tick_top" class="sphx-glr-backref-module-matplotlib-axis sphx-glr-backref-type-py-method"><span class="n">ax2</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">tick_top</span></a><span class="p">()</span>
<span class="k">for</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">n1</span></a><span class="p">):</span>
<a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.int64" title="numpy.int64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">j</span></a> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.argmax.html#numpy.argmax" title="numpy.argmax" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">argmax</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pi_sample</span></a><span class="p">[</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a><span class="p">,</span> <span class="p">:])</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a> <span class="o">=</span> <span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a> <span class="o">-</span> <span class="mf">0.5</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a><span class="p">)</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a> <span class="o">=</span> <span class="p">(</span><a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.int64" title="numpy.int64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">j</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d2</span></a> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">con</span></a> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class"><span class="n">ConnectionPatch</span></a><span class="p">(</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a><span class="o">=</span><a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a><span class="o">=</span><a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a><span class="p">,</span> <span class="n">coordsA</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/transformations.html#matplotlib.transforms.CompositeGenericTransform" title="matplotlib.transforms.CompositeGenericTransform" class="sphx-glr-backref-module-matplotlib-transforms sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax1</span><span class="o">.</span><span class="n">transData</span></a><span class="p">,</span> <span class="n">coordsB</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/transformations.html#matplotlib.transforms.CompositeGenericTransform" title="matplotlib.transforms.CompositeGenericTransform" class="sphx-glr-backref-module-matplotlib-transforms sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax2</span><span class="o">.</span><span class="n">transData</span></a><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.figure.Figure.add_artist.html#matplotlib.figure.Figure.add_artist" title="matplotlib.figure.Figure.add_artist" class="sphx-glr-backref-module-matplotlib-figure sphx-glr-backref-type-py-method"><span class="n">fig</span><span class="o">.</span><span class="n">add_artist</span></a><span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">con</span></a><span class="p">)</span>
<span class="k">for</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">d1</span></a><span class="p">):</span>
<a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.int64" title="numpy.int64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">j</span></a> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.argmax.html#numpy.argmax" title="numpy.argmax" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">argmax</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray" title="numpy.ndarray" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">pi_feature</span></a><span class="p">[</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a><span class="p">,</span> <span class="p">:])</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a> <span class="o">=</span> <span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">i</span></a><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <a href="https://numpy.org/doc/stable/reference/arrays.scalars.html#numpy.int64" title="numpy.int64" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-attribute"><span class="n">j</span></a><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">con</span></a> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class"><span class="n">ConnectionPatch</span></a><span class="p">(</span>
<a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a><span class="o">=</span><a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyA</span></a><span class="p">,</span> <a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a><span class="o">=</span><a href="https://docs.python.org/3/library/stdtypes.html#tuple" title="builtins.tuple" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">xyB</span></a><span class="p">,</span> <span class="n">coordsA</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/transformations.html#matplotlib.transforms.CompositeGenericTransform" title="matplotlib.transforms.CompositeGenericTransform" class="sphx-glr-backref-module-matplotlib-transforms sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax1</span><span class="o">.</span><span class="n">transData</span></a><span class="p">,</span> <span class="n">coordsB</span><span class="o">=</span><a href="https://matplotlib.org/stable/api/transformations.html#matplotlib.transforms.CompositeGenericTransform" title="matplotlib.transforms.CompositeGenericTransform" class="sphx-glr-backref-module-matplotlib-transforms sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ax2</span><span class="o">.</span><span class="n">transData</span></a><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"blue"</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.figure.Figure.add_artist.html#matplotlib.figure.Figure.add_artist" title="matplotlib.figure.Figure.add_artist" class="sphx-glr-backref-module-matplotlib-figure sphx-glr-backref-type-py-method"><span class="n">fig</span><span class="o">.</span><span class="n">add_artist</span></a><span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.ConnectionPatch.html#matplotlib.patches.ConnectionPatch" title="matplotlib.patches.ConnectionPatch" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">con</span></a><span class="p">)</span>
</pre></div>
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<img src="../../_images/sphx_glr_plot_COOT_002.png" srcset="../../_images/sphx_glr_plot_COOT_002.png" alt="Transpose($X_2$)" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>CO-Optimal Transport cost at iteration 1: 0.09599610821598674
CO-Optimal Transport distance = 0.09600
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