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<a class="parent" href="../index.html">transformers</a>
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<a class="parent" href="index.html">gmlp</a>
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<h1>Pay Attention to MLPs (gMLP)</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/2105.08050">Pay Attention to MLPs</a>.</p>
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<p>This paper introduces a Multilayer Perceptron (MLP) based architecture with gating,
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which they name <strong>gMLP</strong>. It consists of a stack of $L$ <em>gMLP</em> blocks.</p>
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<p>Here is <a href="experiment.html">the training code</a> for a gMLP model based autoregressive model.</p>
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<p><a href="https://app.labml.ai/run/01bd941ac74c11eb890c1d9196651a4a"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></a></p>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">21</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span>
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<span class="lineno">22</span>
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<span class="lineno">23</span><span class="kn">import</span> <span class="nn">torch</span>
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<span class="lineno">24</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>gMLP Block</h2>
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<p>Each block does the following transformations to input embeddings
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$X \in \mathbb{R}^{n \times d}$ where $n$ is the sequence length
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and $d$ is the dimensionality of the embeddings:</p>
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<p>
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<script type="math/tex; mode=display">\begin{align}
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Z &= \sigma(XU) \\
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\tilde{Z} &= s(Z) \\
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Y &= \tilde{Z}V \\
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\end{align}</script>
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</p>
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<p>where $V$ and $U$ are learnable projection weights.
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$s(\cdot)$ is the Spacial Gating Unit defined below.
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Output dimensionality of $s(\cdot)$ will be half of $Z$.
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$\sigma$ is an activation function such as
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<a href="https://pytorch.org/docs/stable/generated/torch.nn.GELU.html">GeLU</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">27</span><span class="k">class</span> <span class="nc">GMLPBlock</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<p><code>d_model</code> is the dimensionality ($d$) of $X$
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<code>d_ffn</code> is the dimensionality of $Z$
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<code>seq_len</code> is the length of the token sequence ($n$)</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="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">d_ffn</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">:</span> <span class="nb">int</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">54</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</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-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>Normalization layer fro Pre-Norm</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="bp">self</span><span class="o">.</span><span class="n">norm</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">LayerNorm</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-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>Activation function $\sigma$</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="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">GELU</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>Projection layer for $Z = \sigma(XU)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">60</span> <span class="bp">self</span><span class="o">.</span><span class="n">proj1</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">d_model</span><span class="p">,</span> <span class="n">d_ffn</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>Spacial Gating Unit $s(\cdot)$</p>
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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="bp">self</span><span class="o">.</span><span class="n">sgu</span> <span class="o">=</span> <span class="n">SpacialGatingUnit</span><span class="p">(</span><span class="n">d_ffn</span><span class="p">,</span> <span class="n">seq_len</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>Projection layer for $Y = \tilde{Z}V$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">64</span> <span class="bp">self</span><span class="o">.</span><span class="n">proj2</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">d_ffn</span> <span class="o">//</span> <span class="mi">2</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-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>Embedding size (required by <a href="../models.html#Encoder">Encoder</a>.
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We use the encoder module from transformer architecture and plug
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<em>gMLP</em> block as a replacement for the <a href="../models.html#Encoder">Transformer Layer</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">68</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="n">d_model</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 doc-strings'>
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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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<ul>
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<li><code>x</code> is the input embedding tensor $X$ of shape <code>[seq_len, batch_size, d_model]</code></li>
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<li><code>mask</code> is a boolean mask of shape <code>[seq_len, seq_len, 1]</code> that controls the visibility of tokens
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among each other.</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">70</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="o">*</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> <span class="n">mask</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-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>Keep a copy for shortcut connection</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">77</span> <span class="n">shortcut</span> <span class="o">=</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-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>Normalize $X$</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="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">x</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>Projection and activation $Z = \sigma(XU)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">81</span> <span class="n">z</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">proj1</span><span class="p">(</span><span class="n">x</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>Spacial Gating Unit $\tilde{Z} = s(Z)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">83</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sgu</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">mask</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>Final projection $Y = \tilde{Z}V$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">85</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">proj2</span><span class="p">(</span><span class="n">z</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>Add the shortcut connection</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">return</span> <span class="n">z</span> <span class="o">+</span> <span class="n">shortcut</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 doc-strings'>
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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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<h2>Spatial Gating Unit</h2>
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<p>
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<script type="math/tex; mode=display">s(Z) = Z_1 \odot f_{W,b}(Z_2)</script>
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</p>
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<p>where $f_{W,b}(Z) = W Z + b$ is a linear transformation along the sequence dimension,
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and $\odot$ is element-wise multiplication.
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$Z$ is split into to parts of equal size $Z_1$ and $Z_2$ along the channel dimension (embedding 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">91</span><span class="k">class</span> <span class="nc">SpacialGatingUnit</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-18'>
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<div class='docs doc-strings'>
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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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<ul>
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<li><code>d_z</code> is the dimensionality of $Z$</li>
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<li><code>seq_len</code> is the sequence length</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">101</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_z</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">:</span> <span class="nb">int</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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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">106</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</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>Normalization layer before applying $f_{W,b}(\cdot)$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">108</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">LayerNorm</span><span class="p">([</span><span class="n">d_z</span> <span class="o">//</span> <span class="mi">2</span><span class="p">])</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-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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<p>Weight $W$ in $f_{W,b}(\cdot)$.</p>
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<p>The paper notes that it’s important to initialize weights to small values and the bias to $1$,
|
|
so that during the initial training $s(\cdot)$ is close to identity (apart from the split).</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">113</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight</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">seq_len</span><span class="p">,</span> <span class="n">seq_len</span><span class="p">)</span><span class="o">.</span><span class="n">uniform_</span><span class="p">(</span><span class="o">-</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.01</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-22'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
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</div>
|
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<p>Weight $b$ in $f_{W,b}(\cdot)$</p>
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<p>The paper notes that it’s important to initialize bias to $1$.</p>
|
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</div>
|
|
<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">117</span> <span class="bp">self</span><span class="o">.</span><span class="n">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">ones</span><span class="p">(</span><span class="n">seq_len</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-23'>
|
|
<div class='docs doc-strings'>
|
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<div class='section-link'>
|
|
<a href='#section-23'>#</a>
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</div>
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<ul>
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|
<li><code>z</code> is the input $Z$ of shape <code>[seq_len, batch_size, d_z]</code></li>
|
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<li><code>mask</code> is is a boolean mask of shape <code>[seq_len, seq_len, 1]</code> that controls the visibility of tokens
|
|
among each other. The last dimension of size <code>1</code> is the batch, which we have in other transformer
|
|
implementations and was left for compatibility.</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">119</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">z</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">mask</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-24'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-24'>#</a>
|
|
</div>
|
|
<p>Get sequence length</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">128</span> <span class="n">seq_len</span> <span class="o">=</span> <span class="n">z</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>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-25'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-25'>#</a>
|
|
</div>
|
|
<p>Split $Z$ into $Z_1$ and $Z_2$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">130</span> <span class="n">z1</span><span class="p">,</span> <span class="n">z2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">chunk</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">2</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>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-26'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-26'>#</a>
|
|
</div>
|
|
<p>Check mask</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">133</span> <span class="k">if</span> <span class="n">mask</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</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><code>mask</code> has shape <code>[seq_len_q, seq_len_k, batch_size]</code>.
|
|
The batch dimension should be of size <code>1</code> because this implementation supports
|
|
only same mask for all samples in the batch.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">137</span> <span class="k">assert</span> <span class="n">mask</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">mask</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">seq_len</span>
|
|
<span class="lineno">138</span> <span class="k">assert</span> <span class="n">mask</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">seq_len</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>Here we only support the same mask for all samples</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">140</span> <span class="k">assert</span> <span class="n">mask</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></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>Remove the batch dimension</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">142</span> <span class="n">mask</span> <span class="o">=</span> <span class="n">mask</span><span class="p">[:,</span> <span class="p">:,</span> <span class="mi">0</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>Normalize $Z_2$ before $f_{W,b}(\cdot)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">145</span> <span class="n">z2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">z2</span><span class="p">)</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
|
|
<p>Get the weight matrix; truncate if larger than <code>seq_len</code></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">147</span> <span class="n">weight</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight</span><span class="p">[:</span><span class="n">seq_len</span><span class="p">,</span> <span class="p">:</span><span class="n">seq_len</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-32'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-32'>#</a>
|
|
</div>
|
|
<p>Apply mask to the weights.</p>
|
|
<p>If $W_{i,j}$ is $0$ then $f_{W,b}(Z_2)_i$ will not get any information
|
|
from token $j$.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">152</span> <span class="k">if</span> <span class="n">mask</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
|
<span class="lineno">153</span> <span class="n">weight</span> <span class="o">=</span> <span class="n">weight</span> <span class="o">*</span> <span class="n">mask</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-33'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-33'>#</a>
|
|
</div>
|
|
<p>$f_{W,b}(Z_2) = W Z_2 + b$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">156</span> <span class="n">z2</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">'ij,jbd->ibd'</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">z2</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">bias</span><span class="p">[:</span><span class="n">seq_len</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-34'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-34'>#</a>
|
|
</div>
|
|
<p>$Z_1 \odot f_{W,b}(Z_2)$</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">159</span> <span class="k">return</span> <span class="n">z1</span> <span class="o">*</span> <span class="n">z2</span></pre></div>
|
|
</div>
|
|
</div>
|
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</div>
|
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</div>
|
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