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<p>我们介绍了变分图自动编码器(VGAE)，这是一个基于变分自动编码器(VAE)的图结构数据的无监督学习框架[2，3]。该模型利用了潜在变量，能够学习无向图的可解释的潜在表示(见图1)。</p> |
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<p>我们用一个图形卷积网络(GCN)[4]编码器和一个简单的内积译码器演示了这个模型。我们的模型在引文网络中的链接预测任务上取得了有竞争力的结果。与大多数现有的基于图结构数据的无监督学习和链接预测模型[5，6，7，8]相比，我们的模型可以自然地加入节点特征，从而显著提高了在许多基准数据集上的预测性能。<br> |
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<img src="https://cdn.nlark.com/yuque/0/2021/png/2888751/1626705138908-4d841dc4-4ada-4133-905b-0a64b89e8001.png#align=left&display=inline&height=565&margin=%5Bobject%20Object%5D&name=image.png&originHeight=565&originWidth=643&size=235085&status=done&style=none&width=643" alt="image.png"><br> |
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图1：在CORA引文网络数据集上训练的无监督VGAE模型的潜在空间[1]。<br> |
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灰色线条表示引用链接。颜色表示文档类别(培训期间未提供)。最好在屏幕上观看。</p> |
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<p>**定义 **我们得到一个无向、无权图<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">G</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="script">V</mi><mo separator="true">,</mo><mi mathvariant="script">E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G}=(\mathcal{V}, \mathcal{E})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathcal" style="margin-right:0.08222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mclose">)</span></span></span></span>，其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi mathvariant="script">V</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">N=|\mathcal{V}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathcal" style="margin-right:0.08222em;">V</span><span class="mord">∣</span></span></span></span>个节点。我们引入了<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">G</mi><mtext> 的邻接矩阵</mtext><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathcal{G } \text { 的}\text {邻接矩阵} \mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7833em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">的</span></span><span class="mord text"><span class="mord cjk_fallback">邻接矩阵</span></span><span class="mord mathbf">A</span></span></span></span>(假设对角元素设为1，即每个结点都与其自身相连)和它的次数矩阵<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">D</mi></mrow><annotation encoding="application/x-tex">\mathbf{D}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">D</span></span></span></span>。进一步引入了随机潜变量<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{z}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，将其归纳为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">N \times F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>的矩阵<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Z</mi></mrow><annotation encoding="application/x-tex">\mathbf{Z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">Z</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> 节点特征被归结为 </mtext><mi>N</mi><mo>×</mo><mi>D</mi><mtext> 的矩阵 </mtext><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\text { 节点特征被归结为 } N \times D \text { 的矩阵 } \mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">节点特征被归结为</span><span class="mord"> </span></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">的矩阵</span><span class="mord"> </span></span><span class="mord mathbf">X</span></span></span></span></p> |
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<p>**推理模型 **我们采用一个由两层GCN参数化的简单推理模型：<br> |
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo>∣</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi>q</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo>∣</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mtext> with </mtext><mi>q</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo>∣</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo fence="true">)</mo></mrow><mo>=</mo><mi mathvariant="script">N</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo>∣</mo><msub><mi mathvariant="bold-italic">μ</mi><mi>i</mi></msub><mo separator="true">,</mo><mi mathvariant="normal">diag</mi><mo></mo><mrow><mo fence="true">(</mo><msubsup><mi mathvariant="bold-italic">σ</mi><mi>i</mi><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">q(\mathbf{Z} \mid \mathbf{X}, \mathbf{A})=\prod_{i=1}^{N} q\left(\mathbf{z}_{i} \mid \mathbf{X}, \mathbf{A}\right), \text { with } q\left(\mathbf{z}_{i} \mid \mathbf{X}, \mathbf{A}\right)=\mathcal{N}\left(\mathbf{z}_{i} \mid \boldsymbol{\mu}_{i}, \operatorname{diag}\left(\boldsymbol{\sigma}_{i}^{2}\right)\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord mathbf">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2809em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> with </span></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.01389em;">diag</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">σ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span><br> |
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这里，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">μ</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">G</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">N</mi></mrow><mi mathvariant="bold-italic">μ</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{\mu}=\mathrm{GCN}_{\boldsymbol{\mu}}(\mathbf{X}, \mathbf{A})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">GCN</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">μ</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span></span></span></span>是平均向量<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">μ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{\mu}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">μ</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2175em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span></span></span></span>的矩阵；类似的，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo></mo><mi>σ</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">GCN</mi><mo></mo></mrow><mi>σ</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\log \sigma=\operatorname{GCN}_{\sigma}(\mathbf{X}, \mathbf{A})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop"><span class="mord mathrm">GCN</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span></span></span></span>。<br> |
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这两层<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>C</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">GCN</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">GCN</span></span></span></span>被定义为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">G</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">N</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mi mathvariant="bold">A</mi><mo>~</mo></mover><mi mathvariant="normal">ReLU</mi><mo></mo><mrow><mo fence="true">(</mo><mover accent="true"><mi mathvariant="bold">A</mi><mo>~</mo></mover><mi mathvariant="bold">X</mi><msub><mi mathvariant="bold">W</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><msub><mi mathvariant="bold">W</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{GCN}(\mathbf{X}, \mathbf{A})=\tilde{\mathbf{A}} \operatorname{ReLU}\left(\tilde{\mathbf{A}} \mathbf{X} \mathbf{W}_{0}\right) \mathbf{W}_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">GCN</span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.923em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">A</span></span><span style="top:-3.6051em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mord mathrm">ReLU</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.923em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">A</span></span><span style="top:-3.6051em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，具有权重矩阵<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">W</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="normal">GCN</mi><mo></mo></mrow><mi mathvariant="bold-italic">μ</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo><mtext> and </mtext><msub><mrow><mi mathvariant="normal">G</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">N</mi></mrow><mi mathvariant="bold-italic">σ</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\operatorname{GCN}_{\boldsymbol{\mu}}(\mathbf{X}, \mathbf{A}) \text { and } \mathrm{GCN}_{\boldsymbol{\sigma}}(\mathbf{X}, \mathbf{A})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop"><span class="mop"><span class="mord mathrm">GCN</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight">μ</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span><span class="mord text"><span class="mord"> and </span></span><span class="mord"><span class="mord"><span class="mord mathrm">GCN</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord boldsymbol mtight" style="margin-right:0.03704em;">σ</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span></span></span></span>共享第一层参数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">W</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ReLU</mi><mo></mo><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>=</mo><mi>max</mi><mo></mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mtext> 和 </mtext><mover accent="true"><mi mathvariant="bold">A</mi><mo>~</mo></mover><mo>=</mo><msup><mi mathvariant="bold">D</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><msup><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">D</mi></mrow><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow><annotation encoding="application/x-tex">\operatorname{ReLU}(\cdot)=\max (0, \cdot) \text { 和 } |
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\tilde{\mathbf{A}}=\mathbf{D}^{-\frac{1}{2}} \mathbf{A D}^{-\frac{1}{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">ReLU</span></span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.173em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">⋅</span><span class="mclose">)</span><span class="mord text"><span class="mord"> </span><span class="mord cjk_fallback">和</span><span class="mord"> </span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.923em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">A</span></span><span style="top:-3.6051em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.03em;"></span><span class="mord"><span class="mord mathbf">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">AD</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.03em;"><span style="top:-3.439em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span>是对称正规化邻接矩阵。</p> |
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<p>**生成模型 **我们的生成模型是由潜在变量之间的内积给出的：<br> |
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold">A</mi><mo>∣</mo><mi mathvariant="bold">Z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msubsup><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi>p</mi><mrow><mo fence="true">(</mo><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∣</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">z</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mtext> with </mtext><mi>p</mi><mrow><mo fence="true">(</mo><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>∣</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi mathvariant="bold">z</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><mi>σ</mi><mrow><mo fence="true">(</mo><msubsup><mi mathvariant="bold">z</mi><mi>i</mi><mi mathvariant="normal">⊤</mi></msubsup><msub><mi mathvariant="bold">z</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">p(\mathbf{A} \mid \mathbf{Z})=\prod_{i=1}^{N} \prod_{j=1}^{N} p\left(A_{i j} \mid \mathbf{z}_{i}, \mathbf{z}_{j}\right), \text { with } p\left(A_{i j}=1 \mid \mathbf{z}_{i}, \mathbf{z}_{j}\right)=\sigma\left(\mathbf{z}_{i}^{\top} \mathbf{z}_{j}\right),</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathbf">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.417em;vertical-align:-0.4358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9812em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> with </span></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span><br> |
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其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{i j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span></span></span></span>的元素，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>是Logistic Sigmoid函数。</p> |
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<p>**学习 **我们优化了变分下界<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi><mtext> w.r.t. </mtext></mrow><annotation encoding="application/x-tex">\mathcal{L} \text { w.r.t. }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span><span class="mord text"><span class="mord"> w.r.t. </span></span></span></span></span>。变化参数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">W</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.016em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：<br> |
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi><mo>=</mo><msub><mi mathvariant="double-struck">E</mi><mrow><mi>q</mi><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo>∣</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mi>log</mi><mo></mo><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold">A</mi><mo>∣</mo><mi mathvariant="bold">Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi mathvariant="normal">KL</mi><mo></mo><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo>∣</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∥</mi><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}=\mathbb{E}_{q(\mathbf{Z} \mid \mathbf{X}, \mathbf{A})}[\log p(\mathbf{A} \mid \mathbf{Z})]-\operatorname{KL}[q(\mathbf{Z} \mid \mathbf{X}, \mathbf{A}) \| p(\mathbf{Z})]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathbb">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span><span class="mopen mtight">(</span><span class="mord mathbf mtight">Z</span><span class="mrel mtight">∣</span><span class="mord mathbf mtight">X</span><span class="mpunct mtight">,</span><span class="mord mathbf mtight">A</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathbf">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">Z</span><span class="mclose">)]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">KL</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord mathbf">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span><span class="mord">∥</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathbf">Z</span><span class="mclose">)]</span></span></span></span><br> |
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其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi></mrow><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mi mathvariant="normal">∥</mi><mi>p</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{KL}[q(\cdot) \| p(\cdot)]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">KL</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mord">∥</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)]</span></span></span></span>是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mtext> and </mtext><mi>p</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(\cdot) \text { and } p(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span><span class="mord text"><span class="mord"> and </span></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span>之间的Kullback-Leibler散度。我们进一步取高斯先验<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi mathvariant="bold">Z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∏</mo><mi>i</mi></msub><mi>p</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold">z</mi><mi mathvariant="bold">i</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mo>∏</mo><mi>i</mi></msub><mi mathvariant="script">N</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold">z</mi><mi>i</mi></msub><mo>∣</mo><mn>0</mn><mo separator="true">,</mo><mi mathvariant="bold">I</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">p(\mathbf{Z})=\prod_{i} p\left(\mathbf{z}_{\mathbf{i}}\right)=\prod_{i} \mathcal{N}\left(\mathbf{z}_{i} \mid 0, \mathbf{I}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathbf">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0497em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0497em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.14736em;">N</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathbf">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">I</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>。对于非常稀疏的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span></span></span></span>，对<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">L</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">L</span></span></span></span>中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A_{i j}=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>的项或<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A_{i j}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>的子采样项重新加权可能是有益的。我们选择前者进行下面的实验。我们执行全批次梯度下降，并利用重新参数化技巧[2]进行训练。对于一种无特征的方法，我们只需放弃对<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">X</span></span></span></span>的依赖，而用GCN中的单位矩阵替换<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">X</span></span></span></span>。</p> |
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<p>**非概率图自动编码器(GAE)模型 **对于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">V</mi><mi mathvariant="bold">G</mi><mi mathvariant="bold">A</mi><mi mathvariant="bold">E</mi></mrow><annotation encoding="application/x-tex">\mathbf{VGAE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord mathbf">VGAE</span></span></span></span></span>模型的非概率变体，我们计算嵌入<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Z</mi></mrow><annotation encoding="application/x-tex">\mathbf{Z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">Z</span></span></span></span>和重构的邻接矩阵<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">A</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\mathbf{A}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9495em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9495em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">A</span></span><span style="top:-3.2551em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span>，如下所示：<br> |
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">A</mi><mo>^</mo></mover><mo>=</mo><mi>σ</mi><mrow><mo fence="true">(</mo><mi mathvariant="bold">Z</mi><msup><mi mathvariant="bold">Z</mi><mi mathvariant="normal">⊤</mi></msup><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mtext> with </mtext><mspace width="1em"/><mi mathvariant="bold">Z</mi><mo>=</mo><mi mathvariant="normal">GCN</mi><mo></mo><mo stretchy="false">(</mo><mi mathvariant="bold">X</mi><mo separator="true">,</mo><mi mathvariant="bold">A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat{\mathbf{A}}=\sigma\left(\mathbf{Z} \mathbf{Z}^{\top}\right), \text { with } \quad \mathbf{Z}=\operatorname{GCN}(\mathbf{X}, \mathbf{A})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9495em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9495em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathbf">A</span></span><span style="top:-3.2551em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathbf">Z</span><span class="mord"><span class="mord mathbf">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord"> with </span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathbf">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mord mathrm">GCN</span></span><span class="mopen">(</span><span class="mord mathbf">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbf">A</span><span class="mclose">)</span></span></span></span></p> |
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<h1>2 关于链接预测的实验</h1> |
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<p>我们在几个流行的引文网络数据上展示了VGAE和GAE模型在链接预测任务中学习有意义的潜在嵌入的能力[1]。这些模型是在这些数据集的不完整版本上训练的，其中部分引文链接(边)已被移除，而所有节点特征都被保留。我们从先前移除的边和相同数量的随机采样的未连接节点对(非边)形成验证和测试集。</p> |
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<p>我们根据模型正确区分有边和无边的能力对模型进行比较。验证和测试集分别包含5%和10%的引文链接。验证集用于优化超参数。我们比较了两个流行的基准：频谱聚类(SC)[5]和DeepWalk(DW)[6]。SC和DW都提供了节点嵌入<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Z</mi></mrow><annotation encoding="application/x-tex">\mathbf{Z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">Z</span></span></span></span>。我们使用公式4(左侧)计算重建邻接矩阵元素的分数。由于性能相当，我们省略了DW[7，8]的最新变体。SC和DW都不支持输入功能。</p> |
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<p>对于VGAE和GAE，我们按照[9]中所述初始化权重。我们使用Adam[10]训练了200次迭代，学习率为0.01。我们在所有的实验中都使用了32维的隐藏层和16维的潜变量。对于SC，我们使用[11]中的实现，嵌入维数为128。对于DW，我们使用[8]的作者提供的实现，使用他们论文中使用的标准设置，即嵌入维数为128，每个节点10个长度为80的随机游动，上下文大小为10，针对单个纪元进行训练。</p> |
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<p>**讨论 **引文网络中链接预测任务的结果汇总在表1中。GAE<em>和VGAE</em>表示不使用输入特征的实验，GAE和VGAE使用输入特征。我们报告测试集上每个模型的ROC曲线下面积(AUC)和平均精度(AP)分数。数字显示了对固定数据集拆分进行随机初始化的10次运行的平均结果和标准误差。</p> |
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<p>表1：引文网络中的链接预测任务。有关数据集的详细信息，请参见[1]。<br> |
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<img src="https://cdn.nlark.com/yuque/0/2021/png/2888751/1626708100508-5c0b0382-18a7-4cdf-8058-e18d7067b083.png#align=left&display=inline&height=208&margin=%5Bobject%20Object%5D&name=image.png&originHeight=208&originWidth=794&size=44248&status=done&style=none&width=794" alt="image.png"><br> |
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VGAE和GAE在这项平淡无奇的任务上都取得了有竞争力的结果。添加输入功能可显著提高数据集的预测性能。与内积解码器结合使用时，高斯先验可能是一个糟糕的选择，因为后者试图将嵌入推离零中心(参见图1)。然而，VGAE模型在CORA和Citeseer数据集上都实现了更高的预测性能。</p> |
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<p>未来的工作将研究更适合的先验分布，更灵活的生成模型，以及随机梯度下降算法的应用，以提高可扩展性。</p> |
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<p>References<br> |
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[1] P. Sen, G. M. Namata, M. Bilgic, L. Getoor, B. Gallagher, and T. Eliassi-Rad. Collective<br> |
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classification in network data. AI Magazine, 29(3):93–106, 2008.<br> |
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[2] D. P. Kingma and M. Welling.<br> |
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Auto-encoding variational bayes.<br> |
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In Proceedings of the<br> |
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International Conference on Learning Representations (ICLR), 2014.<br> |
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[3] D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate<br> |
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inference in deep generative models. In Proceedings of The 31st International Conference on<br> |
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Machine Learning (ICML), 2014.<br> |
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[4] T. N. Kipf and M. Welling. Semi-supervised classification with graph convolutional networks.<br> |
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arXiv preprint arXiv:1609.02907, 2016.<br> |
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[5] L. Tang and H. Liu. Leveraging social media networks for classification. Data Mining and<br> |
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Knowledge Discovery, 23(3):447–478, 2011.<br> |
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[6] B. Perozzi, R. Al-Rfou, and S. Skiena. Deepwalk: Online learning of social representations. In<br> |
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Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and<br> |
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Data Mining (KDD), pages 701–710. ACM, 2014.<br> |
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[7] J. Tang, M. Qu, M. Wang, M. Zhang, J. Yan, and Q. Mei. Line: Large-scale information network<br> |
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embedding. In Proceedings of the 24th International Conference on World Wide Web, pages<br> |
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1067–1077. ACM, 2015.<br> |
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[8] A. Grover and J. Leskovec. node2vec: Scalable feature learning for networks. In Proceedings<br> |
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of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining<br> |
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(KDD), 2016.<br> |
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[9] X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural<br> |
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networks. In Aistats, volume 9, pages 249–256, 2010.<br> |
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[10] D. P. Kingma and J. L. Ba. Adam: A method for stochastic optimization. In Proceedings of the<br> |
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International Conference on Learning Representations (ICLR), 2015.<br> |
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[11] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel,<br> |
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P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher,<br> |
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M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine<br> |
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Learning Research, 12:2825–2830, 2011.</p> |
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class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-text">1 一种图结构数据的潜变量模型</span></a></li><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-text">2 关于链接预测的实验</span></a></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="fas fa-history"></i><span>最新文章</span></div><div class="aside-list"><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2024/07/06/git%E5%B1%80%E5%9F%9F%E7%BD%91%E9%83%A8%E7%BD%B2%E6%96%B9%E6%B3%95/" title="git局域网部署方法">git局域网部署方法</a><time datetime="2024-07-06T08:13:10.000Z" title="发表于 2024-07-06 16:13:10">2024-07-06</time></div></div><div class="aside-list-item no-cover"><div class="content"><a class="title" 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region: '', |
|
|
urls: [window.location.pathname], |
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|
includeReply: false |
|
|
}).then(function (res) { |
|
|
countELement.innerText = res[0].count |
|
|
}).catch(function (err) { |
|
|
console.error(err); |
|
|
}); |
|
|
} |
|
|
|
|
|
const runFn = () => { |
|
|
init() |
|
|
|
|
|
} |
|
|
|
|
|
const loadTwikoo = () => { |
|
|
if (typeof twikoo === 'object') { |
|
|
setTimeout(runFn,0) |
|
|
return |
|
|
} |
|
|
getScript('https://cdn.jsdelivr.net/npm/twikoo/dist/twikoo.all.min.js').then(runFn) |
|
|
} |
|
|
|
|
|
if ('Twikoo' === 'Twikoo' || !false) { |
|
|
if (false) btf.loadComment(document.getElementById('twikoo-wrap'), loadTwikoo) |
|
|
else loadTwikoo() |
|
|
} else { |
|
|
window.loadOtherComment = () => { |
|
|
loadTwikoo() |
|
|
} |
|
|
} |
|
|
})()</script><script>function addGitalkSource () { |
|
|
const ele = document.createElement('link') |
|
|
ele.rel = 'stylesheet' |
|
|
ele.href= 'https://cdn.jsdelivr.net/npm/gitalk/dist/gitalk.min.css' |
|
|
document.getElementsByTagName('head')[0].appendChild(ele) |
|
|
} |
|
|
|
|
|
function loadGitalk () { |
|
|
function initGitalk () { |
|
|
var gitalk = new Gitalk(Object.assign({ |
|
|
clientID: 'f9bab17720653ab07b39', |
|
|
clientSecret: 'e77fbcac5c4043ddf8d845aee7d6d5f118f607e2', |
|
|
repo: 'ruoneo-comment-repo', |
|
|
owner: 'ruoneo', |
|
|
admin: ['ruoneo'], |
|
|
id: '58257cb378d3c80c6d76e2931da26a76', |
|
|
updateCountCallback: commentCount |
|
|
},null)) |
|
|
|
|
|
gitalk.render('gitalk-container') |
|
|
} |
|
|
|
|
|
if (typeof Gitalk === 'function') initGitalk() |
|
|
else { |
|
|
addGitalkSource() |
|
|
getScript('https://cdn.jsdelivr.net/npm/gitalk/dist/gitalk.min.js').then(initGitalk) |
|
|
} |
|
|
} |
|
|
|
|
|
function commentCount(n){ |
|
|
let isCommentCount = document.querySelector('#post-meta .gitalk-comment-count') |
|
|
if (isCommentCount) { |
|
|
isCommentCount.innerHTML= n |
|
|
} |
|
|
} |
|
|
|
|
|
if ('Twikoo' === 'Gitalk' || !false) { |
|
|
if (false) btf.loadComment(document.getElementById('gitalk-container'), loadGitalk) |
|
|
else loadGitalk() |
|
|
} else { |
|
|
function loadOtherComment () { |
|
|
loadGitalk() |
|
|
} |
|
|
}</script></div></div></body></html> |
