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<h1>Source code for ot.factored</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Factored OT solvers (low rank, cost or OT plan)</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">.utils</span><span class="w"> </span><span class="kn">import</span> <span class="n">dist</span><span class="p">,</span> <span class="n">get_lowrank_lazytensor</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">.lp</span><span class="w"> </span><span class="kn">import</span> <span class="n">emd</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">.bregman</span><span class="w"> </span><span class="kn">import</span> <span class="n">sinkhorn</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"factored_optimal_transport"</span><span class="p">]</span>
<div class="viewcode-block" id="factored_optimal_transport">
<a class="viewcode-back" href="../../gen_modules/ot.factored.html#ot.factored_optimal_transport">[docs]</a>
<span class="k">def</span><span class="w"> </span><span class="nf">factored_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">reg</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span>
<span class="n">r</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">X0</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">stopThr</span><span class="o">=</span><span class="mf">1e-7</span><span class="p">,</span>
<span class="n">numItermax</span><span class="o">=</span><span class="mi">100</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="o">**</span><span class="n">kwargs</span><span class="p">,</span>
<span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Solves factored OT problem and return OT plans and intermediate distribution</span>
<span class="sd"> This function solve the following OT problem [40]_</span>
<span class="sd"> .. math::</span>
<span class="sd"> \mathop{\arg \min}_\mu \quad W_2^2(\mu_a,\mu)+ W_2^2(\mu,\mu_b)</span>
<span class="sd"> where :</span>
<span class="sd"> - :math:`\mu_a` and :math:`\mu_b` are empirical distributions.</span>
<span class="sd"> - :math:`\mu` is an empirical distribution with r samples</span>
<span class="sd"> And returns the two OT plans between</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 in</span>
<span class="sd"> :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"> numItermax : int, optional</span>
<span class="sd"> Max number of iterations</span>
<span class="sd"> stopThr : float, optional</span>
<span class="sd"> Stop threshold on the relative 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"> Ga: array-like, shape (ns, r)</span>
<span class="sd"> Optimal transportation matrix between source and the intermediate</span>
<span class="sd"> distribution</span>
<span class="sd"> Gb: array-like, shape (r, nt)</span>
<span class="sd"> Optimal transportation matrix between the intermediate and target</span>
<span class="sd"> distribution</span>
<span class="sd"> X: array-like, shape (r, d)</span>
<span class="sd"> Support of the intermediate distribution</span>
<span class="sd"> log: dict, optional</span>
<span class="sd"> If input log is true, a dictionary containing the cost and dual</span>
<span class="sd"> variables and exit status</span>
<span class="sd"> .. _references-factored:</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [40] Forrow, A., Hütter, J. C., Nitzan, M., Rigollet, P., Schiebinger,</span>
<span class="sd"> G., & Weed, J. (2019, April). Statistical optimal transport via factored</span>
<span class="sd"> couplings. In The 22nd International Conference on Artificial</span>
<span class="sd"> Intelligence and Statistics (pp. 2454-2465). PMLR.</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">n_a</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="n">n_b</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="n">d</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">1</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">a</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n_a</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="o">/</span> <span class="n">n_a</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">b</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n_b</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="o">/</span> <span class="n">n_b</span>
<span class="k">if</span> <span class="n">X0</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">d</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="k">else</span><span class="p">:</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X0</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">r</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="o">/</span> <span class="n">r</span>
<span class="k">def</span><span class="w"> </span><span class="nf">solve_ot</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">,</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">):</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">dist</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="n">X2</span><span class="p">)</span>
<span class="k">if</span> <span class="n">reg</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">G</span><span class="p">,</span> <span class="n">log</span> <span class="o">=</span> <span class="n">sinkhorn</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">reg</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="kc">True</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">"cost"</span><span class="p">]</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">G</span> <span class="o">*</span> <span class="n">M</span><span class="p">)</span>
<span class="k">return</span> <span class="n">G</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">emd</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="n">norm_delta</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># solve the barycenter</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numItermax</span><span class="p">):</span>
<span class="n">old_X</span> <span class="o">=</span> <span class="n">X</span>
<span class="c1"># solve OT with template</span>
<span class="n">Ga</span><span class="p">,</span> <span class="n">loga</span> <span class="o">=</span> <span class="n">solve_ot</span><span class="p">(</span><span class="n">Xa</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
<span class="n">Gb</span><span class="p">,</span> <span class="n">logb</span> <span class="o">=</span> <span class="n">solve_ot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Xb</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Ga</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">Xa</span><span class="p">)</span> <span class="o">+</span> <span class="n">nx</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Gb</span><span class="p">,</span> <span class="n">Xb</span><span class="p">))</span> <span class="o">*</span> <span class="n">r</span>
<span class="n">delta</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">old_X</span><span class="p">)</span>
<span class="k">if</span> <span class="n">delta</span> <span class="o"><</span> <span class="n">stopThr</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">if</span> <span class="n">log</span><span class="p">:</span>
<span class="n">norm_delta</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">delta</span><span class="p">)</span>
<span class="k">if</span> <span class="n">log</span><span class="p">:</span>
<span class="n">log_dic</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"delta_iter"</span><span class="p">:</span> <span class="n">norm_delta</span><span class="p">,</span>
<span class="s2">"ua"</span><span class="p">:</span> <span class="n">loga</span><span class="p">[</span><span class="s2">"u"</span><span class="p">],</span>
<span class="s2">"va"</span><span class="p">:</span> <span class="n">loga</span><span class="p">[</span><span class="s2">"v"</span><span class="p">],</span>
<span class="s2">"ub"</span><span class="p">:</span> <span class="n">logb</span><span class="p">[</span><span class="s2">"u"</span><span class="p">],</span>
<span class="s2">"vb"</span><span class="p">:</span> <span class="n">logb</span><span class="p">[</span><span class="s2">"v"</span><span class="p">],</span>
<span class="s2">"costa"</span><span class="p">:</span> <span class="n">loga</span><span class="p">[</span><span class="s2">"cost"</span><span class="p">],</span>
<span class="s2">"costb"</span><span class="p">:</span> <span class="n">logb</span><span class="p">[</span><span class="s2">"cost"</span><span class="p">],</span>
<span class="s2">"lazy_plan"</span><span class="p">:</span> <span class="n">get_lowrank_lazytensor</span><span class="p">(</span><span class="n">Ga</span> <span class="o">*</span> <span class="n">r</span><span class="p">,</span> <span class="n">Gb</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="n">nx</span><span class="p">),</span>
<span class="p">}</span>
<span class="k">return</span> <span class="n">Ga</span><span class="p">,</span> <span class="n">Gb</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">log_dic</span>
<span class="k">return</span> <span class="n">Ga</span><span class="p">,</span> <span class="n">Gb</span><span class="p">,</span> <span class="n">X</span></div>
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