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<h1>FNet: Mixing Tokens with Fourier Transforms</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of the paper <a href="https://papers.labml.ai/paper/2105.03824">FNet: Mixing Tokens with Fourier Transforms</a>.</p>
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<p>This paper replaces the <a href="../mha.html">self-attention layer</a> with two <a href="https://en.wikipedia.org/wiki/Discrete_Fourier_transform">Fourier transforms</a> to <em>mix</em> tokens. This is a <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">7</span><span class="mord">×</span></span></span></span> more efficient than self-attention. The accuracy loss of using this over self-attention is about 92% for <a href="https://paperswithcode.com/method/bert">BERT</a> on <a href="https://paperswithcode.com/dataset/glue">GLUE benchmark</a>.</p>
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<h2>Mixing tokens with two Fourier transforms</h2>
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<p>We apply Fourier transform along the hidden dimension (embedding dimension) and then along the sequence dimension.</p>
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<p><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqb" style=""><span class="mord coloredeq eqi" style=""><span class="mord mathcal" style="">R</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mclose" style="">)</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></p>
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<p>where <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span></span></span></span> is the embedding input, <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span></span></span></span> stands for the fourier transform and <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqi" style=""><span class="mord mathcal" style="">R</span></span></span></span></span> stands for the real component in complex numbers.</p>
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<p>This is very simple to implement on PyTorch - just 1 line of code. The paper suggests using a precomputed DFT matrix and doing matrix multiplication to get the Fourier transformation.</p>
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<p>Here is <a href="experiment.html">the training code</a> for using a FNet based model for classifying <a href="https://paperswithcode.com/dataset/ag-news">AG News</a>.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">41</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span>
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<span class="lineno">42</span>
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<span class="lineno">43</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">44</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>FNet - Mix tokens</h2>
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<p>This module simply implements <span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqb" style=""><span class="mord coloredeq eqi" style=""><span class="mord mathcal" style="">R</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mclose" style="">)</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span></p>
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<p>The structure of this module is made similar to a <a href="../mha.html">standard attention module</a> so that we can simply replace it.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">47</span><span class="k">class</span> <span class="nc">FNetMix</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<p> The <a href="../mha.html">normal attention module</a> can be fed with different token embeddings for <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span></span></span></span>,<span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span></span></span></span>, and <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span> and a mask.</p>
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<p>We follow the same function signature so that we can replace it directly.</p>
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<p>For FNet mixing, <span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span></span> and masking is not possible. Shape of <code class="highlight"><span></span><span class="n">query</span></code>
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(and <code class="highlight"><span></span><span class="n">key</span></code>
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and <code class="highlight"><span></span><span class="n">value</span></code>
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) is <code class="highlight"><span></span><span class="p">[</span><span class="n">seq_len</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">d_model</span><span class="p">]</span></code>
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.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">60</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">value</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">mask</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-3'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-3'>#</a>
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</div>
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<p><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqf" style=""><span class="mord text" style=""><span class="mord" style="">query</span></span></span></span></span></span>,<span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqj" style=""><span class="mord text" style=""><span class="mord" style="">key</span></span></span></span></span></span>, and <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqg" style=""><span class="mord text" style=""><span class="mord" style="">value</span></span></span></span></span></span> all should be equal to <span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span></span></span></span> for token mixing </p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">72</span> <span class="k">assert</span> <span class="n">query</span> <span class="ow">is</span> <span class="n">key</span> <span class="ow">and</span> <span class="n">key</span> <span class="ow">is</span> <span class="n">value</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<p>Token mixing doesn't support masking. i.e. all tokens will see all other token embeddings. </p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">74</span> <span class="k">assert</span> <span class="n">mask</span> <span class="ow">is</span> <span class="kc">None</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-5'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-5'>#</a>
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</div>
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<p>Assign to <code class="highlight"><span></span><span class="n">x</span></code>
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for clarity </p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">77</span> <span class="n">x</span> <span class="o">=</span> <span class="n">query</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-6'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-6'>#</a>
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</div>
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<p>Apply the Fourier transform along the hidden (embedding) dimension <span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>
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<p>The output of the Fourier transform is a tensor of <a href="https://pytorch.org/docs/stable/complex_numbers.html">complex numbers</a>. </p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">84</span> <span class="n">fft_hidden</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-7'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-7'>#</a>
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</div>
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<p>Apply the Fourier transform along the sequence dimension <span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mclose" style="">)</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span> </p>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">87</span> <span class="n">fft_seq</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">fft_hidden</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span></pre></div>
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</div>
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<div class='section' id='section-8'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-8'>#</a>
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</div>
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<p>Get the real component <span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="mord coloredeq eqb" style=""><span class="mord coloredeq eqi" style=""><span class="mord mathcal" style="">R</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">seq</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">(</span></span></span><span class="mord coloredeq eqe" style=""><span class="mord" style=""><span class="mord coloredeq eqh" style=""><span class="mord mathcal" style="margin-right:0.09931em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord text mtight" style=""><span class="mord mtight" style="">hidden</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord coloredeq eql" style=""><span class="mord mathnormal" style="">x</span></span><span class="mclose" style="">)</span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span><span class="mord" style=""><span class="delimsizing size1" style=""><span style="">)</span></span></span></span></span></span></span></span> </p>
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</div>
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<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">91</span> <span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">fft_seq</span><span class="p">)</span></pre></div>
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