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<h1>Source code for ot.weak</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Weak optimal ransport solvers</span>
<span class="sd">"""</span>
<span class="c1"># Author: Remi Flamary <remi.flamary@polytehnique.edu></span>
<span class="c1">#</span>
<span class="c1"># License: MIT License</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">.backend</span><span class="w"> </span><span class="kn">import</span> <span class="n">get_backend</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">.optim</span><span class="w"> </span><span class="kn">import</span> <span class="n">cg</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="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"weak_optimal_transport"</span><span class="p">]</span>
<div class="viewcode-block" id="weak_optimal_transport">
<a class="viewcode-back" href="../../gen_modules/ot.weak.html#ot.weak_optimal_transport">[docs]</a>
<span class="k">def</span><span class="w"> </span><span class="nf">weak_optimal_transport</span><span class="p">(</span>
<span class="n">Xa</span><span class="p">,</span> <span class="n">Xb</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">G0</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Solves the weak optimal transport problem between two empirical distributions</span>
<span class="sd"> .. math::</span>
<span class="sd"> \gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2</span>
<span class="sd"> s.t. \ \gamma \mathbf{1} = \mathbf{a}</span>
<span class="sd"> \gamma^T \mathbf{1} = \mathbf{b}</span>
<span class="sd"> \gamma \geq 0</span>
<span class="sd"> where :</span>
<span class="sd"> - :math:`X^a` and :math:`X^b` are the sample matrices.</span>
<span class="sd"> - :math:`\mathbf{a}` and :math:`\mathbf{b}` are the sample weights</span>
<span class="sd"> .. note:: This function is backend-compatible and will work on arrays</span>
<span class="sd"> from all compatible backends. But the algorithm uses the C++ CPU backend</span>
<span class="sd"> which can lead to copy overhead on GPU arrays.</span>
<span class="sd"> Uses the conditional gradient algorithm to solve the problem proposed</span>
<span class="sd"> in :ref:`[39] <references-weak>`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> Xa : (ns,d) array-like, float</span>
<span class="sd"> Source samples</span>
<span class="sd"> Xb : (nt,d) array-like, float</span>
<span class="sd"> Target samples</span>
<span class="sd"> a : (ns,) array-like, float</span>
<span class="sd"> Source histogram (uniform weight if empty list)</span>
<span class="sd"> b : (nt,) array-like, float</span>
<span class="sd"> Target histogram (uniform weight if empty list))</span>
<span class="sd"> G0 : (ns,nt) array-like, float</span>
<span class="sd"> initial guess (default is indep joint density)</span>
<span class="sd"> numItermax : int, optional</span>
<span class="sd"> Max number of iterations</span>
<span class="sd"> numItermaxEmd : int, optional</span>
<span class="sd"> Max number of iterations for emd</span>
<span class="sd"> stopThr : float, optional</span>
<span class="sd"> Stop threshold on the relative variation (>0)</span>
<span class="sd"> stopThr2 : float, optional</span>
<span class="sd"> Stop threshold on the absolute variation (>0)</span>
<span class="sd"> verbose : bool, optional</span>
<span class="sd"> Print information along iterations</span>
<span class="sd"> log : bool, optional</span>
<span class="sd"> record log if True</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> gamma: array-like, shape (ns, nt)</span>
<span class="sd"> Optimal transportation matrix for the given</span>
<span class="sd"> parameters</span>
<span class="sd"> log: dict, optional</span>
<span class="sd"> If input log is true, a dictionary containing the</span>
<span class="sd"> cost and dual variables and exit status</span>
<span class="sd"> .. _references-weak:</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [39] Gozlan, N., Roberto, C., Samson, P. M., & Tetali, P. (2017).</span>
<span class="sd"> Kantorovich duality for general transport costs and applications.</span>
<span class="sd"> Journal of Functional Analysis, 273(11), 3327-3405.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> ot.bregman.sinkhorn : Entropic regularized OT</span>
<span class="sd"> ot.optim.cg : General regularized OT</span>
<span class="sd"> """</span>
<span class="n">nx</span> <span class="o">=</span> <span class="n">get_backend</span><span class="p">(</span><span class="n">Xa</span><span class="p">,</span> <span class="n">Xb</span><span class="p">)</span>
<span class="n">Xa2</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">(</span><span class="n">Xa</span><span class="p">)</span>
<span class="n">Xb2</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">(</span><span class="n">Xb</span><span class="p">)</span>
<span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">a2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">Xa</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span> <span class="o">/</span> <span class="n">Xa</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">a2</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">b2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">Xb</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span> <span class="o">/</span> <span class="n">Xb</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">b2</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="c1"># init uniform</span>
<span class="k">if</span> <span class="n">G0</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">T0</span> <span class="o">=</span> <span class="n">a2</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">b2</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">T0</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">to_numpy</span><span class="p">(</span><span class="n">G0</span><span class="p">)</span>
<span class="c1"># weak OT loss</span>
<span class="k">def</span><span class="w"> </span><span class="nf">f</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">Xa2</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">Xb2</span><span class="p">)</span> <span class="o">/</span> <span class="n">a2</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="c1"># weak OT gradient</span>
<span class="k">def</span><span class="w"> </span><span class="nf">df</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Xa2</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">Xb2</span><span class="p">)</span> <span class="o">/</span> <span class="n">a2</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">],</span> <span class="n">Xb2</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="c1"># solve with conditional gradient and return solution</span>
<span class="k">if</span> <span class="n">log</span><span class="p">:</span>
<span class="n">res</span><span class="p">,</span> <span class="n">log</span> <span class="o">=</span> <span class="n">cg</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">T0</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="n">log</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="n">log</span><span class="p">[</span><span class="s2">"u"</span><span class="p">]</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">log</span><span class="p">[</span><span class="s2">"u"</span><span class="p">],</span> <span class="n">type_as</span><span class="o">=</span><span class="n">Xa</span><span class="p">)</span>
<span class="n">log</span><span class="p">[</span><span class="s2">"v"</span><span class="p">]</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">log</span><span class="p">[</span><span class="s2">"v"</span><span class="p">],</span> <span class="n">type_as</span><span class="o">=</span><span class="n">Xb</span><span class="p">)</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">type_as</span><span class="o">=</span><span class="n">Xa</span><span class="p">),</span> <span class="n">log</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">from_numpy</span><span class="p">(</span>
<span class="n">cg</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">T0</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="n">log</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="n">type_as</span><span class="o">=</span><span class="n">Xa</span>
<span class="p">)</span></div>
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