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<div class="section" id="torch-einsum">
<h1>torch.einsum<a class="headerlink" href="#torch-einsum" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="torch.einsum">
<code class="sig-prename descclassname">torch.</code><code class="sig-name descname">einsum</code><span class="sig-paren">(</span><em class="sig-param">equation</em>, <em class="sig-param">*operands</em><span class="sig-paren">)</span> → Tensor<a class="reference internal" href="../_modules/torch/functional.html#einsum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.einsum" title="Permalink to this definition">¶</a></dt>
<dd><p>This function provides a way of computing multilinear expressions (i.e. sums of products) using the
Einstein summation convention.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>equation</strong> (<em>string</em>) – The equation is given in terms of lower case letters (indices) to be associated
with each dimension of the operands and result. The left hand side lists the operands
dimensions, separated by commas. There should be one index letter per tensor dimension.
The right hand side follows after <cite>-></cite> and gives the indices for the output.
If the <cite>-></cite> and right hand side are omitted, it implicitly defined as the alphabetically
sorted list of all indices appearing exactly once in the left hand side.
The indices not apprearing in the output are summed over after multiplying the operands
entries.
If an index appears several times for the same operand, a diagonal is taken.
Ellipses <cite>…</cite> represent a fixed number of dimensions. If the right hand side is inferred,
the ellipsis dimensions are at the beginning of the output.</p></li>
<li><p><strong>operands</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – The operands to compute the Einstein sum of.</p></li>
</ul>
</dd>
</dl>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function does not optimize the given expression, so a different formula for the same computation may
run faster or consume less memory. Projects like opt_einsum (<a class="reference external" href="https://optimized-einsum.readthedocs.io/en/stable/">https://optimized-einsum.readthedocs.io/en/stable/</a>)
can optimize the formula for you.</p>
</div>
<p>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'i,j->ij'</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="c1"># outer product</span>
<span class="go">tensor([[-0.0570, -0.0286, -0.0231, 0.0197],</span>
<span class="go"> [ 1.2616, 0.6335, 0.5113, -0.4351],</span>
<span class="go"> [ 1.4452, 0.7257, 0.5857, -0.4984],</span>
<span class="go"> [-0.4647, -0.2333, -0.1883, 0.1603],</span>
<span class="go"> [-1.1130, -0.5588, -0.4510, 0.3838]])</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">l</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">r</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'bn,anm,bm->ba'</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="c1"># compare torch.nn.functional.bilinear</span>
<span class="go">tensor([[-0.3430, -5.2405, 0.4494],</span>
<span class="go"> [ 0.3311, 5.5201, -3.0356]])</span>
<span class="gp">>>> </span><span class="n">As</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Bs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'bij,bjk->bik'</span><span class="p">,</span> <span class="n">As</span><span class="p">,</span> <span class="n">Bs</span><span class="p">)</span> <span class="c1"># batch matrix multiplication</span>
<span class="go">tensor([[[-1.0564, -1.5904, 3.2023, 3.1271],</span>
<span class="go"> [-1.6706, -0.8097, -0.8025, -2.1183]],</span>
<span class="go"> [[ 4.2239, 0.3107, -0.5756, -0.2354],</span>
<span class="go"> [-1.4558, -0.3460, 1.5087, -0.8530]],</span>
<span class="go"> [[ 2.8153, 1.8787, -4.3839, -1.2112],</span>
<span class="go"> [ 0.3728, -2.1131, 0.0921, 0.8305]]])</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ii->i'</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span> <span class="c1"># diagonal</span>
<span class="go">tensor([-0.7825, 0.8291, -0.1936])</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'...ii->...i'</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span> <span class="c1"># batch diagonal</span>
<span class="go">tensor([[-1.0864, 0.7292, 0.0569],</span>
<span class="go"> [-0.9725, -1.0270, 0.6493],</span>
<span class="go"> [ 0.5832, -1.1716, -1.5084],</span>
<span class="go"> [ 0.4041, -1.1690, 0.8570]])</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</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="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'...ij->...ji'</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span> <span class="c1"># batch permute</span>
<span class="go">torch.Size([2, 3, 5, 4])</span>
</pre></div>
</div>
</dd></dl>
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