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<h1>Relative Multi-Headed Attention</h1>
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<p>This is an implementation of
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<a href="https://arxiv.org/abs/1901.02860">Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context</a>.</p>
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<p>Transformer has a limited attention span,
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equal to the length of the sequence trained in parallel.
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All these positions have a fixed positional encoding.
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Transformer XL increases this attention span by letting
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each of the positions pay attention to precalculated past embeddings.
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For instance if the context length is $l$ it will keep the embeddings of
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all layers for previous batch of length $l$ and feed them to current step.
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If we use fixed-positional encodings these pre-calculated embeddings will have
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the same positions as the current context.
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They introduce relative positional encoding, where the positional encodings
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are introduced at the attention calculation.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">27</span><span></span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">28</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
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<span class="lineno">29</span>
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<span class="lineno">30</span><span class="kn">from</span> <span class="nn">labml.logger</span> <span class="kn">import</span> <span class="n">inspect</span>
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<span class="lineno">31</span><span class="kn">from</span> <span class="nn">labml_nn.transformers.mha</span> <span class="kn">import</span> <span class="n">MultiHeadAttention</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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<p>This method shifts $i^{th}$ row of a matrix by $i$ columns.</p>
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<p>If the input is <code>[[1, 2 ,3], [4, 5 ,6], [7, 8, 9]]</code>, the shifted
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result would be <code>[[1, 2 ,3], [0, 4, 5], [9, 0, 7]]</code>.
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<em>Ideally we should mask out the lower triangle but it’s ok for our purpose</em>.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">34</span><span class="k">def</span> <span class="nf">shift_right</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></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'>
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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>Concatenate a column of zeros</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">44</span> <span class="n">zero_pad</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">new_zeros</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="mi">1</span><span class="p">,</span> <span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
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<span class="lineno">45</span> <span class="n">x_padded</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cat</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">zero_pad</span><span class="p">],</span> <span class="n">dim</span><span class="o">=</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-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>Remove excess elements from the end</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">x_padded</span> <span class="o">=</span> <span class="n">x_padded</span><span class="o">.</span><span class="n">view</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">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</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">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
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<span class="lineno">49</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x_padded</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">view_as</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="lineno">50</span>
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<span class="lineno">51</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 doc-strings'>
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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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<h2>Relative Multi-Head Attention Module</h2>
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<p>We override <a href="mha.html">Multi-Head Attention</a> module so we only need to
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write the <code>get_scores</code> method.</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="k">class</span> <span class="nc">RelativeMultiHeadAttention</span><span class="p">(</span><span class="n">MultiHeadAttention</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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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">62</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">heads</span><span class="p">:</span> <span class="nb">int</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="o">=</span> <span class="mf">0.1</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-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>The linear transformations doesn’t need a bias since we take care of it when
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calculating scores.
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However having a bias for <code>value</code> might make sense.</p>
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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="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">heads</span><span class="p">,</span> <span class="n">d_model</span><span class="p">,</span> <span class="n">dropout_prob</span><span class="p">,</span> <span class="n">bias</span><span class="o">=</span><span class="kc">False</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>Number of relative positions</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">69</span> <span class="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">**</span> <span class="mi">12</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>Relative positional embeddings for key relative to the query.
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We need $2P$ embeddings because the keys can be before or after the query.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">73</span> <span class="bp">self</span><span class="o">.</span><span class="n">key_pos_embeddings</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">*</span> <span class="mi">2</span><span class="p">,</span> <span class="n">heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">d_k</span><span class="p">)),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</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>Relative positional embedding bias for key relative to the query.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">75</span> <span class="bp">self</span><span class="o">.</span><span class="n">key_pos_bias</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">*</span> <span class="mi">2</span><span class="p">,</span> <span class="n">heads</span><span class="p">)),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</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>Positional embeddings for the query is independent of the position of the query</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="bp">self</span><span class="o">.</span><span class="n">query_pos_bias</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">d_k</span><span class="p">)),</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</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 doc-strings'>
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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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<h3>Get relative attention scores</h3>
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<p>With absolute attention</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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A^{abs}_{j} &= lin_q(X^q_i + P_i)^\top lin_k(X^k_j + P_j) \\
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&= \underset{\color{lightgreen}{A}}{Q_i^\top K_j} +
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\underset{\color{lightgreen}{B}}{Q_i^\top U^K_j} +
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\underset{\color{lightgreen}{C}}{{U^Q_i}^\top K_j} +
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\underset{\color{lightgreen}{D}}{{U^Q_i}^\top U^K_j}
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\end{align}</script>
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</p>
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<p>where $Q_i, K_j$, are linear transformations of
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original embeddings $X^q_i, X^k_j$
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and $U^Q_i, U^K_j$ are linear transformations of
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absolute positional encodings $P_i, P_j$.</p>
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<p>They reason out that the attention to a given key should be the same regardless of
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the position of query.
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Hence replace $\underset{\color{lightgreen}{C}}{{U^Q_i}^\top K_j}$
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with a constant $\underset{\color{lightgreen}{C}}{\color{orange}{v^\top} K_j}$.</p>
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<p>For the second and third terms relative positional encodings are introduced.
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So $\underset{\color{lightgreen}{B}}{Q_i^\top U^K_j}$ is
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replaced with $\underset{\color{lightgreen}{B}}{Q_i^\top \color{orange}{R_{i - j}}}$
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and $\underset{\color{lightgreen}{D}}{{U^Q_i}^\top U^K_j}$
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with $\underset{\color{lightgreen}{D}}{\color{orange}{S_{i-j}}}$.</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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A^{rel}_{i,j} &= \underset{\mathbf{\color{lightgreen}{A}}}{Q_i^\top K_j} +
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\underset{\mathbf{\color{lightgreen}{B}}}{Q_i^\top \color{orange}{R_{i - j}}} +
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\underset{\mathbf{\color{lightgreen}{C}}}{\color{orange}{v^\top} K_j} +
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\underset{\mathbf{\color{lightgreen}{D}}}{\color{orange}{S_{i-j}}}
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\end{align}</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">79</span> <span class="k">def</span> <span class="nf">get_scores</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></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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<p>$\color{orange}{R_k}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">118</span> <span class="n">key_pos_emb</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key_pos_embeddings</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">-</span> <span class="n">query</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="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">+</span> <span class="n">key</span><span class="o">.</span><span class="n">shape</span><span class="p">[</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-13'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-13'>#</a>
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</div>
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<p>$\color{orange}{S_k}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">120</span> <span class="n">key_pos_bias</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key_pos_bias</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">-</span> <span class="n">query</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="bp">self</span><span class="o">.</span><span class="n">P</span> <span class="o">+</span> <span class="n">key</span><span class="o">.</span><span class="n">shape</span><span class="p">[</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-14'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-14'>#</a>
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</div>
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<p>$\color{orange}{v^\top}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">122</span> <span class="n">query_pos_bias</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">query_pos_bias</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="p">:,</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-15'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-15'>#</a>
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</div>
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<p>${(\color{lightgreen}{\mathbf{A + C}})}_{i,j} =
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Q_i^\top K_j +
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\color{orange}{v^\top} K_jZ$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">127</span> <span class="n">ac</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ibhd,jbhd->ijbh'</span><span class="p">,</span> <span class="n">query</span> <span class="o">+</span> <span class="n">query_pos_bias</span><span class="p">,</span> <span class="n">key</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-16'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-16'>#</a>
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</div>
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<p>$\color{lightgreen}{\mathbf{B’}_{i,k}} = Q_i^\top \color{orange}{R_k}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">129</span> <span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ibhd,jhd->ijbh'</span><span class="p">,</span> <span class="n">query</span><span class="p">,</span> <span class="n">key_pos_emb</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-17'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-17'>#</a>
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</div>
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<p>$\color{lightgreen}{\mathbf{D’}_{i,k}} = \color{orange}{S_k}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">131</span> <span class="n">d</span> <span class="o">=</span> <span class="n">key_pos_bias</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:,</span> <span class="kc">None</span><span class="p">,</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-18'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-18'>#</a>
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</div>
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<p>Shift the rows of $\color{lightgreen}{\mathbf{(B’ + D’)}_{i,k}}$
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to get <script type="math/tex; mode=display">\color{lightgreen}{\mathbf{(B + D)}_{i,j} = \mathbf{(B' + D')}_{i,i - j}}</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">134</span> <span class="n">bd</span> <span class="o">=</span> <span class="n">shift_right</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">d</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-19'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-19'>#</a>
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</div>
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<p>Remove extra positions</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">136</span> <span class="n">bd</span> <span class="o">=</span> <span class="n">bd</span><span class="p">[:,</span> <span class="o">-</span><span class="n">key</span><span class="o">.</span><span class="n">shape</span><span class="p">[</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-20'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-20'>#</a>
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</div>
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<p>Return the sum <script type="math/tex; mode=display">
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\underset{\mathbf{\color{lightgreen}{A}}}{Q_i^\top K_j} +
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\underset{\mathbf{\color{lightgreen}{B}}}{Q_i^\top \color{orange}{R_{i - j}}} +
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\underset{\mathbf{\color{lightgreen}{C}}}{\color{orange}{v^\top} K_j} +
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\underset{\mathbf{\color{lightgreen}{D}}}{\color{orange}{S_{i-j}}}
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</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">144</span> <span class="k">return</span> <span class="n">ac</span> <span class="o">+</span> <span class="n">bd</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-21'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-21'>#</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">147</span><span class="k">def</span> <span class="nf">_test_shift_right</span><span class="p">():</span>
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<span class="lineno">148</span> <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</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="mi">3</span><span class="p">],</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="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
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<span class="lineno">149</span> <span class="n">inspect</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="lineno">150</span> <span class="n">inspect</span><span class="p">(</span><span class="n">shift_right</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
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<span class="lineno">151</span>
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<span class="lineno">152</span> <span class="n">x</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">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
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<span class="lineno">153</span> <span class="n">inspect</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<span class="lineno">154</span> <span class="n">inspect</span><span class="p">(</span><span class="n">shift_right</span><span class="p">(</span><span class="n">x</span><span class="p">)[:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<span class="lineno">155</span>
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<span class="lineno">156</span> <span class="n">x</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">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
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<span class="lineno">157</span> <span class="n">inspect</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<span class="lineno">158</span> <span class="n">inspect</span><span class="p">(</span><span class="n">shift_right</span><span class="p">(</span><span class="n">x</span><span class="p">)[:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<span class="lineno">159</span>
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<span class="lineno">160</span>
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<span class="lineno">161</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">162</span> <span class="n">_test_shift_right</span><span class="p">()</span></pre></div>
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</div>
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</div>
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</div>
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</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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<script type="text/x-mathjax-config">
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"HTML-CSS": { fonts: ["TeX"] }
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