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<h1>Fixed Positional Encodings</h1>
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<p>The positional encoding encodes the position along the sequence into
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a vector of size <code>d_model</code>.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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PE_{p,2i} &= sin\Bigg(\frac{p}{10000^{\frac{2i}{d_{model}}}}\Bigg) \\
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PE_{p,2i + 1} &= cos\Bigg(\frac{p}{10000^{\frac{2i}{d_{model}}}}\Bigg)
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\end{align}</script>
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</p>
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<p>Where $1 \leq 2i, 2i + 1 \leq d_{model}$
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are the feature indexes in the encoding, and $p$ is the position.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">23</span><span></span><span class="kn">import</span> <span class="nn">math</span>
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<span class="lineno">24</span>
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<span class="lineno">25</span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="lineno">26</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">27</span><span class="kn">import</span> <span class="nn">torch.nn</span> <span class="k">as</span> <span class="nn">nn</span>
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<span class="lineno">28</span>
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<span class="lineno">29</span><span class="kn">from</span> <span class="nn">labml_helpers.module</span> <span class="kn">import</span> <span class="n">Module</span></pre></div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">32</span><span class="k">class</span> <span class="nc">PositionalEncoding</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span></pre></div>
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</div>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">33</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d_model</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">dropout_prob</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">max_len</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">5000</span><span class="p">):</span>
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<span class="lineno">34</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
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<span class="lineno">35</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="n">dropout_prob</span><span class="p">)</span>
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<span class="lineno">36</span>
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<span class="lineno">37</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'positional_encodings'</span><span class="p">,</span> <span class="n">get_positional_encoding</span><span class="p">(</span><span class="n">d_model</span><span class="p">,</span> <span class="n">max_len</span><span class="p">),</span> <span class="kc">False</span><span class="p">)</span></pre></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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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">39</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">x</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span>
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<span class="lineno">40</span> <span class="n">pe</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">positional_encodings</span><span class="p">[:</span><span class="n">x</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">detach</span><span class="p">()</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
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<span class="lineno">41</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">pe</span>
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<span class="lineno">42</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dropout</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="lineno">43</span> <span class="k">return</span> <span class="n">x</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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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">46</span><span class="k">def</span> <span class="nf">get_positional_encoding</span><span class="p">(</span><span class="n">d_model</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">max_len</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">5000</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-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>Empty encodings vectors</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">48</span> <span class="n">encodings</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">max_len</span><span class="p">,</span> <span class="n">d_model</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-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>Position indexes</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">50</span> <span class="n">position</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_len</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">1</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>$2 * i$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">52</span> <span class="n">two_i</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</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>$10000^{\frac{2i}{d_{model}}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">54</span> <span class="n">div_term</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">two_i</span> <span class="o">*</span> <span class="o">-</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mf">10000.0</span><span class="p">)</span> <span class="o">/</span> <span class="n">d_model</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-9'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-9'>#</a>
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</div>
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<p>$PE_{p,2i} = sin\Bigg(\frac{p}{10000^{\frac{2i}{d_{model}}}}\Bigg)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">56</span> <span class="n">encodings</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">position</span> <span class="o">*</span> <span class="n">div_term</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-10'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-10'>#</a>
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</div>
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<p>$PE_{p,2i + 1} = cos\Bigg(\frac{p}{10000^{\frac{2i}{d_{model}}}}\Bigg)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">58</span> <span class="n">encodings</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">position</span> <span class="o">*</span> <span class="n">div_term</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-11'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-11'>#</a>
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</div>
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<p>Add batch dimension</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">61</span> <span class="n">encodings</span> <span class="o">=</span> <span class="n">encodings</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
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<span class="lineno">62</span>
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<span class="lineno">63</span> <span class="k">return</span> <span class="n">encodings</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-12'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-12'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">66</span><span class="k">def</span> <span class="nf">_test_positional_encoding</span><span class="p">():</span>
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<span class="lineno">67</span> <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
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<span class="lineno">68</span>
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<span class="lineno">69</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
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<span class="lineno">70</span> <span class="n">pe</span> <span class="o">=</span> <span class="n">get_positional_encoding</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
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<span class="lineno">71</span> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">100</span><span class="p">),</span> <span class="n">pe</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span><span class="mi">8</span><span class="p">]</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
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<span class="lineno">72</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s2">"dim </span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]])</span>
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<span class="lineno">73</span> <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Positional encoding"</span><span class="p">)</span>
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<span class="lineno">74</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<span class="lineno">75</span>
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<span class="lineno">76</span>
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<span class="lineno">77</span><span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
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<span class="lineno">78</span> <span class="n">_test_positional_encoding</span><span class="p">()</span></pre></div>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
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<!-- MathJax configuration -->
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tex2jax: {
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},
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// Center justify equations in code and markdown cells. Elsewhere
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