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<h1>Graph Attention Networks (GAT)</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://papers.labml.ai/paper/1710.10903">Graph Attention Networks</a>.</p>
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<p>GATs work on graph data.
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A graph consists of nodes and edges connecting nodes.
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For example, in Cora dataset the nodes are research papers and the edges are citations that
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connect the papers.</p>
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<p>GAT uses masked self-attention, kind of similar to <a href="../../transformers/mha.html">transformers</a>.
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GAT consists of graph attention layers stacked on top of each other.
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Each graph attention layer gets node embeddings as inputs and outputs transformed embeddings.
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The node embeddings pay attention to the embeddings of other nodes it’s connected to.
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The details of graph attention layers are included alongside the implementation.</p>
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<p>Here is <a href="experiment.html">the training code</a> for training
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a two-layer GAT on Cora dataset.</p>
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<p><a href="https://app.labml.ai/run/d6c636cadf3511eba2f1e707f612f95d"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></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">30</span><span></span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">31</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">32</span>
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<span class="lineno">33</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>
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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>Graph attention layer</h2>
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<p>This is a single graph attention layer.
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A GAT is made up of multiple such layers.</p>
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<p>It takes
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<script type="math/tex; mode=display">\mathbf{h} = \{ \overrightarrow{h_1}, \overrightarrow{h_2}, \dots, \overrightarrow{h_N} \}</script>,
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where $\overrightarrow{h_i} \in \mathbb{R}^F$ as input
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and outputs
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<script type="math/tex; mode=display">\mathbf{h'} = \{ \overrightarrow{h'_1}, \overrightarrow{h'_2}, \dots, \overrightarrow{h'_N} \}</script>,
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where $\overrightarrow{h’_i} \in \mathbb{R}^{F’}$.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">36</span><span class="k">class</span> <span class="nc">GraphAttentionLayer</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>
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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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<ul>
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<li><code>in_features</code>, $F$, is the number of input features per node</li>
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<li><code>out_features</code>, $F’$, is the number of output features per node</li>
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<li><code>n_heads</code>, $K$, is the number of attention heads</li>
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<li><code>is_concat</code> whether the multi-head results should be concatenated or averaged</li>
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<li><code>dropout</code> is the dropout probability</li>
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<li><code>leaky_relu_negative_slope</code> is the negative slope for leaky relu activation</li>
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</ul>
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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="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">in_features</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">out_features</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">n_heads</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
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<span class="lineno">51</span> <span class="n">is_concat</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span>
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<span class="lineno">52</span> <span class="n">dropout</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.6</span><span class="p">,</span>
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<span class="lineno">53</span> <span class="n">leaky_relu_negative_slope</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.2</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-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">62</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">63</span>
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<span class="lineno">64</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_concat</span> <span class="o">=</span> <span class="n">is_concat</span>
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<span class="lineno">65</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span> <span class="o">=</span> <span class="n">n_heads</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>Calculate the number of dimensions per head</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">68</span> <span class="k">if</span> <span class="n">is_concat</span><span class="p">:</span>
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<span class="lineno">69</span> <span class="k">assert</span> <span class="n">out_features</span> <span class="o">%</span> <span class="n">n_heads</span> <span class="o">==</span> <span class="mi">0</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>If we are concatenating the multiple heads</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">71</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">=</span> <span class="n">out_features</span> <span class="o">//</span> <span class="n">n_heads</span>
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<span class="lineno">72</span> <span class="k">else</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>If we are averaging the multiple heads</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="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">=</span> <span class="n">out_features</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>Linear layer for initial transformation;
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i.e. to transform the node embeddings before self-attention</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">78</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="n">in_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">*</span> <span class="n">n_heads</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-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>Linear layer to compute attention score $e_{ij}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">80</span> <span class="bp">self</span><span class="o">.</span><span class="n">attn</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span> <span class="o">*</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</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-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>The activation for attention score $e_{ij}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">82</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">(</span><span class="n">negative_slope</span><span class="o">=</span><span class="n">leaky_relu_negative_slope</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>Softmax to compute attention $\alpha_{ij}$</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="bp">self</span><span class="o">.</span><span class="n">softmax</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Softmax</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-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>Dropout layer to be applied for attention</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">86</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</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 doc-strings'>
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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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<ul>
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<li><code>h</code>, $\mathbf{h}$ is the input node embeddings of shape <code>[n_nodes, in_features]</code>.</li>
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<li><code>adj_mat</code> is the adjacency matrix of shape <code>[n_nodes, n_nodes, n_heads]</code>.
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We use shape <code>[n_nodes, n_nodes, 1]</code> since the adjacency is the same for each head.</li>
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</ul>
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<p>Adjacency matrix represent the edges (or connections) among nodes.
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<code>adj_mat[i][j]</code> is <code>True</code> if there is an edge from node <code>i</code> to node <code>j</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">88</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">h</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">adj_mat</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-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>Number of nodes</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">99</span> <span class="n">n_nodes</span> <span class="o">=</span> <span class="n">h</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>The initial transformation,
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<script type="math/tex; mode=display">\overrightarrow{g^k_i} = \mathbf{W}^k \overrightarrow{h_i}</script>
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for each head.
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We do single linear transformation and then split it up for each head.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">104</span> <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">linear</span><span class="p">(</span><span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</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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<h4>Calculate attention score</h4>
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<p>We calculate these for each head $k$. <em>We have omitted $\cdot^k$ for simplicity</em>.</p>
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<p>
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<script type="math/tex; mode=display">e_{ij} = a(\mathbf{W} \overrightarrow{h_i}, \mathbf{W} \overrightarrow{h_j}) =
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a(\overrightarrow{g_i}, \overrightarrow{g_j})</script>
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</p>
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<p>$e_{ij}$ is the attention score (importance) from node $j$ to node $i$.
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We calculate this for each head.</p>
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<p>$a$ is the attention mechanism, that calculates the attention score.
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The paper concatenates
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$\overrightarrow{g_i}$, $\overrightarrow{g_j}$
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and does a linear transformation with a weight vector $\mathbf{a} \in \mathbb{R}^{2 F’}$
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followed by a $\text{LeakyReLU}$.</p>
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<p>
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<script type="math/tex; mode=display">e_{ij} = \text{LeakyReLU} \Big(
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\mathbf{a}^\top \Big[
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\overrightarrow{g_i} \Vert \overrightarrow{g_j}
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\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></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>First we calculate
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$\Big[\overrightarrow{g_i} \Vert \overrightarrow{g_j} \Big]$
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for all pairs of $i, j$.</p>
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<p><code>g_repeat</code> gets
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<script type="math/tex; mode=display">\{\overrightarrow{g_1}, \overrightarrow{g_2}, \dots, \overrightarrow{g_N},
|
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\overrightarrow{g_1}, \overrightarrow{g_2}, \dots, \overrightarrow{g_N}, ...\}</script>
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where each node embedding is repeated <code>n_nodes</code> times.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">135</span> <span class="n">g_repeat</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="mi">1</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-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><code>g_repeat_interleave</code> gets
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<script type="math/tex; mode=display">\{\overrightarrow{g_1}, \overrightarrow{g_1}, \dots, \overrightarrow{g_1},
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\overrightarrow{g_2}, \overrightarrow{g_2}, \dots, \overrightarrow{g_2}, ...\}</script>
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where each node embedding is repeated <code>n_nodes</code> times.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">140</span> <span class="n">g_repeat_interleave</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">repeat_interleave</span><span class="p">(</span><span class="n">n_nodes</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-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>Now we concatenate to get
|
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<script type="math/tex; mode=display">\{\overrightarrow{g_1} \Vert \overrightarrow{g_1},
|
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\overrightarrow{g_1}, \Vert \overrightarrow{g_2},
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\dots, \overrightarrow{g_1} \Vert \overrightarrow{g_N},
|
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\overrightarrow{g_2} \Vert \overrightarrow{g_1},
|
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\overrightarrow{g_2}, \Vert \overrightarrow{g_2},
|
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\dots, \overrightarrow{g_2} \Vert \overrightarrow{g_N}, ...\}</script>
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</p>
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</div>
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<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">148</span> <span class="n">g_concat</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">g_repeat_interleave</span><span class="p">,</span> <span class="n">g_repeat</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-19'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-19'>#</a>
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|
</div>
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<p>Reshape so that <code>g_concat[i, j]</code> is $\overrightarrow{g_i} \Vert \overrightarrow{g_j}$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">150</span> <span class="n">g_concat</span> <span class="o">=</span> <span class="n">g_concat</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-20'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-20'>#</a>
|
|
</div>
|
|
<p>Calculate
|
|
<script type="math/tex; mode=display">e_{ij} = \text{LeakyReLU} \Big(
|
|
\mathbf{a}^\top \Big[
|
|
\overrightarrow{g_i} \Vert \overrightarrow{g_j}
|
|
\Big] \Big)</script>
|
|
<code>e</code> is of shape <code>[n_nodes, n_nodes, n_heads, 1]</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">158</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">attn</span><span class="p">(</span><span class="n">g_concat</span><span class="p">))</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>Remove the last dimension of size <code>1</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">160</span> <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
|
|
</div>
|
|
<p>The adjacency matrix should have shape
|
|
<code>[n_nodes, n_nodes, n_heads]</code> or<code>[n_nodes, n_nodes, 1]</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">164</span> <span class="k">assert</span> <span class="n">adj_mat</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="mi">1</span> <span class="ow">or</span> <span class="n">adj_mat</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">n_nodes</span>
|
|
<span class="lineno">165</span> <span class="k">assert</span> <span class="n">adj_mat</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="ow">or</span> <span class="n">adj_mat</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="n">n_nodes</span>
|
|
<span class="lineno">166</span> <span class="k">assert</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">adj_mat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-23'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-23'>#</a>
|
|
</div>
|
|
<p>Mask $e_{ij}$ based on adjacency matrix.
|
|
$e_{ij}$ is set to $- \infty$ if there is no edge from $i$ to $j$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">169</span> <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">masked_fill</span><span class="p">(</span><span class="n">adj_mat</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'-inf'</span><span class="p">))</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-24'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-24'>#</a>
|
|
</div>
|
|
<p>We then normalize attention scores (or coefficients)
|
|
<script type="math/tex; mode=display">\alpha_{ij} = \text{softmax}_j(e_{ij}) =
|
|
\frac{\exp(e_{ij})}{\sum_{j \in \mathcal{N}_i} \exp(e_{ij})}</script>
|
|
</p>
|
|
<p>where $\mathcal{N}_i$ is the set of nodes connected to $i$.</p>
|
|
<p>We do this by setting unconnected $e_{ij}$ to $- \infty$ which
|
|
makes $\exp(e_{ij}) \sim 0$ for unconnected pairs.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">179</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">e</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-25'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-25'>#</a>
|
|
</div>
|
|
<p>Apply dropout regularization</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">182</span> <span class="n">a</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">a</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-26'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-26'>#</a>
|
|
</div>
|
|
<p>Calculate final output for each head
|
|
<script type="math/tex; mode=display">\overrightarrow{h'^k_i} = \sum_{j \in \mathcal{N}_i} \alpha^k_{ij} \overrightarrow{g^k_j}</script>
|
|
</p>
|
|
<p><em>Note:</em> The paper includes the final activation $\sigma$ in $\overrightarrow{h_i}$
|
|
We have omitted this from the Graph Attention Layer implementation
|
|
and use it on the GAT model to match with how other PyTorch modules are defined -
|
|
activation as a separate layer.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">191</span> <span class="n">attn_res</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">'ijh,jhf->ihf'</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-27'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-27'>#</a>
|
|
</div>
|
|
<p>Concatenate the heads</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">194</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_concat</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\overrightarrow{h'_i} = \Bigg\Vert_{k=1}^{K} \overrightarrow{h'^k_i}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">196</span> <span class="k">return</span> <span class="n">attn_res</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_nodes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_heads</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>Take the mean of the heads</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">198</span> <span class="k">else</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-30'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-30'>#</a>
|
|
</div>
|
|
<p>
|
|
<script type="math/tex; mode=display">\overrightarrow{h'_i} = \frac{1}{K} \sum_{k=1}^{K} \overrightarrow{h'^k_i}</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">200</span> <span class="k">return</span> <span class="n">attn_res</span><span class="o">.</span><span class="n">mean</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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