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<title>FNet: Mixing Tokens with Fourier Transforms</title>
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<a class="parent" href="../index.html">transformers</a>
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<a class="parent" href="index.html">fnet</a>
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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
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<a href="https://arxiv.org/abs/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
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<a href="https://en.wikipedia.org/wiki/Discrete_Fourier_transform">Fourier transforms</a> to
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<em>mix</em> tokens.
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This is a $7 \times$ more efficient than self-attention.
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The accuracy loss of using this over self-attention is about 92% for
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<a href="https://paperswithcode.com/method/bert">BERT</a> on
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<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)
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and then along the sequence dimension.</p>
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<p>
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<script type="math/tex; mode=display">
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\mathcal{R}\big(\mathcal{F}_\text{seq} \big(\mathcal{F}_\text{hidden} (x) \big) \big)
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</script>
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</p>
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<p>where $x$ is the embedding input, $\mathcal{F}$ stands for the fourier transform and
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$\mathcal{R}$ 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.
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The paper suggests using a precomputed DFT matrix and doing matrix multiplication to get the
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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
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<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
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<script type="math/tex; mode=display">
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\mathcal{R}\big(\mathcal{F}_\text{seq} \big(\mathcal{F}_\text{hidden} (x) \big) \big)
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</script>
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</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
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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
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$\text{query}$,$\text{key}$, and $\text{value}$ 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, <script type="math/tex; mode=display">x = \text{query} = \text{key} = \text{value}</script> and masking is not possible.
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Shape of <code>query</code> (and <code>key</code> and <code>value</code>) is <code>[seq_len, batch_size, d_model]</code>.</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>$\text{query}$,$\text{key}$, and $\text{value}$ all should be equal to $x$ 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>x</code> 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
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<script type="math/tex; mode=display">\mathcal{F}_\text{hidden} (x)</script>
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</p>
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<p>The output of the Fourier transform is a tensor of
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<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
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<script type="math/tex; mode=display">\mathcal{F}_\text{seq} \big(\mathcal{F}_\text{hidden} (x) \big)</script>
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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">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>
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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
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<script type="math/tex; mode=display">\mathcal{R}\big(\mathcal{F}_\text{seq} \big(\mathcal{F}_\text{hidden} (x) \big) \big)</script>
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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">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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