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‎Paper/VAE/i/index.html‎

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‎Paper/VAE/image-1.png‎

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‎Paper/数学基础/Diffusion model/i/index.html‎

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基本目标
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<h1>Diffusion model</h1>
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<h2 id="_1">基本目标<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h2>
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<p>实现生成的图像分布与现实世界的图像分布尽可能接近</p>
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<p>如何衡量尽可能接近？</p>
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<p>使用最大似然估计</p>
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<p>假设训练模型产生的图像分布为<span class="arithmatex">\(P_{\theta}(x)\)</span>，现实世界的图像分布为<span class="arithmatex">\(P_{data}(x)\)</span></p>
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<p>从<span class="arithmatex">\(P_{data}(x)\)</span>中采样<span class="arithmatex">\(N\)</span>个样本，记为<span class="arithmatex">\(x^1,x^2,\cdots,x^m\)</span></p>
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<div class="arithmatex">\[\begin{align}
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\theta^*=\arg\max_{\theta}\sum_{i=1}^m\log P_{\theta}(x^i)\\
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\approx \arg\max_{\theta}E_{x\sim P_{data}(x)}\log P_{\theta}(x)\\
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=\argmax_{\theta}\int_{x}p_{data}(x)\log P_{\theta}(x)dx\\
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=\argmax_{\theta}\int_{x}p_{data}(x)\log P_{\theta}(x)-\int_{x}p_{data}(x)\log P_{data}(x)dx\\
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= \arg\max_{\theta}\int_{x}p_{data}(x)\frac{\log P_{\theta}(x)}{\log P_{data}(x)}dx\\
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=\argmax KL(p_{\theta}(x)\Vert p_{data}(x))
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\end{align}\]</div>
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<h2 id="fdpforward-diffusion-process">FDP(Forward Diffusion Process)<a class="headerlink" href="#fdpforward-diffusion-process" title="Permanent link">&para;</a></h2>
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<p><span class="arithmatex">\(x_t\)</span>可以直接根据<span class="arithmatex">\(x_0\)</span>和<span class="arithmatex">\(t\)</span>推导出来</p>
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<div class="arithmatex">\[x_t=\sqrt{\prod_{i=1}^{t}\alpha_i}x_0+\sqrt{1-\prod_{i=1}^{t}\alpha_i}\epsilon\]</div>

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‎课程笔记/index.html‎

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‎课程笔记/大一/高等数学/高等数学查漏补缺版/index.html‎

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‎课程笔记/大三上/密码学基础/RSA算法/index.html‎

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