机器学习 中，变分自编码器（Variational Autoencoder，VAE） 是由Diederik P. Kingma和Max Welling提出的一种人工神经网络 结构，属于概率图模式 和变分贝叶斯方法 。[ 1]
VAE与自编码器 模型有关，因为两者在结构上有一定亲和力，但在目标和数学表述上有很大区别。VAE属于概率生成模型（Probabilistic Generative Model），神经网络仅是其中的一个组件，依照功能的不同又可分为编码器和解码器。编码器可将输入变量映射到与变分分布的参数相对应的潜空间（Latent Space），这样便可以产生多个遵循同一分布的不同样本。解码器的功能基本相反，是从潜空间映射回输入空间，以生成数据点。虽然噪声模型的方差可以单独学习而来，但它们通常都是用重参数化技巧（Reparameterization Trick）来训练的。
此类模型最初是为无监督学习 设计的，[ 2] [ 3] 半监督学习 [ 4] [ 5] 监督学习 中也表现出卓越的有效性。[ 6]
VAE是一个分别具有先验和噪声分布的生成模型，一般用最大期望算法（Expectation-Maximization meta-algorithm）来训练。这样可以优化数据似然的下限，用其它方法很难实现这点，且需要q分布或变分后验。这些q分布通常在一个单独的优化过程中为每个单独数据点设定参数；而VAE则用神经网络作为一种摊销手段来联合优化各个数据点，将数据点本身作为输入，输出变分分布的参数。从一个已知的输入空间映射到低维潜空间，这是一种编码过程，因此这张神经网络也叫“编码器”。
解码器则从潜空间映射回输入空间，如作为噪声分布的平均值。也可以用另一个映射到方差的神经网络，为简单起见一般都省略掉了。这时，方差可以用梯度下降法进行优化。
优化模型常用的两个术语是“重构误差（reconstruction error）”和“KL散度 ”。它们都来自概率模型的自由能表达式（Free Energy Expression ），因而根据噪声分布和数据的假定先验而有所不同。例如，像IMAGENET这样的标准VAE任务一般都假设具有高斯分布噪声，但二值化的MNIST这样的任务则需要伯努利噪声。自由能表达式中的KL散度使得与p分布重叠的q分布的概率质量最大化，但这样可能导致出现搜寻模态（Mode-Seeking Behaviour）。自由能表达式的剩余部分是“重构”项，需要用采样逼近来计算其期望。[ 7]
VAE的基本框架。模型接受
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从建立概率模型的角度来看，人们希望用他们选择的参数化概率分布
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{\displaystyle p_{\theta }(x)=p(x|\theta )}
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边缘化 找到
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{\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,}
其中，
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{\displaystyle p_{\theta }({x,z})}
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联合分布 ，和在潜空间中的形式（也就是编码后的
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连锁法则 ，方程可以改写为
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{\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz}
在香草VAE中，通常认为
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{\displaystyle p_{\theta }({x|z})}
高斯分布 。那么
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先验
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{\displaystyle p_{\theta }(z)}
似然值
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{\displaystyle p_{\theta }(x|z)}
后验
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{\displaystyle p_{\theta }(z|x)}
不幸的是，对
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{\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})}
其中
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这样，问题就变成了找到一个好的概率自编码器，其中条件似然分布
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{\displaystyle p_{\theta }(x|z)}
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下面将编码器参数化为
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如同每个深度学习 问题，为了通过反向传播算法 更新神经网络的权重，需要定义一个可微损失函数。
对于VAE，这一思想可以实现为联合优化生成模型参数
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均方误差 和交叉熵 。
作为两个分布之间的距离损失，反向KL散度
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{\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))}
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{\displaystyle q_{\phi }({z|x})}
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{\displaystyle p_{\theta }(z|x)}
[ 8] [ 9]
刚刚定义的距离损失可扩展为
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{\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right]\end{aligned}}}
现在定义证据下界（Evidence lower bound，ELBO）：
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{\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))}
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{\displaystyle \theta ^{*},\phi ^{*}={\underset {\theta ,\phi }{\operatorname {argmax} }}\,L_{\theta ,\phi }(x)}
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{\displaystyle \ln p_{\theta }(x)}
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{\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))}
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{\displaystyle q_{\phi }(\cdot |x)}
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{\displaystyle p_{\theta }(\cdot |x)}
给出的形式不大方便进行最大化，可以用下面的等价形式：
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{\displaystyle L_{\theta ,\phi }(x)=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln p_{\theta }(x|z)\right]-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }(\cdot ))}
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{\displaystyle \ln p_{\theta }(x|z)}
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{\displaystyle z\sim {\mathcal {(}}0,I)}
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{\displaystyle L_{\theta ,\phi }(x)=-{\frac {1}{2}}\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\|x-D_{\theta }(z)\|_{2}^{2}\right]-{\frac {1}{2}}\left(N\sigma _{\phi }(x)^{2}+\|E_{\phi }(x)\|_{2}^{2}-2N\ln \sigma _{\phi }(x)\right)+Const}
重参数化技巧方案。随机变量
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有效搜索到
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{\displaystyle \theta ^{*},\phi ^{*}={\underset {\theta ,\phi }{\operatorname {argmax} }}\,L_{\theta ,\phi }(x)}
梯度下降法 。
它可以很直接地找到
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{\displaystyle \nabla _{\theta }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\nabla _{\theta }\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]}
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{\displaystyle \nabla _{\phi }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]}
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重参数化技巧 （也被称为随机反向传播[ 10] [ 8] [ 11] [ 12]
最重要的例子是当
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重参数化技巧之后的VAE方案 可以通过让
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{\displaystyle {\boldsymbol {\varepsilon }}\sim {\mathcal {N}}(0,{\boldsymbol {I}})}
随机数生成 器”来实现重参数化，并将
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{\displaystyle z=\mu _{\phi }(x)+L_{\phi }(x)\epsilon }
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科列斯基分解 得到：
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{\displaystyle \nabla _{\phi }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\mathbb {E} _{\epsilon }\left[\nabla _{\phi }\ln {\frac {p_{\theta }(x,\mu _{\phi }(x)+L_{\phi }(x)\epsilon )}{q_{\phi }(\mu _{\phi }(x)+L_{\phi }(x)\epsilon |x)}}\right]}
随机梯度下降法 了。
由于我们重参数化了
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{\displaystyle \ln q_{\phi }(z|x)=\ln q_{0}(\epsilon )-\ln |\det(\partial _{\epsilon }z)|}
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许多VAE的应用和扩展已被用来使其适应其他领域，并提升性能。
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[ 13] [ 14]
条件性VAE（CVAE）在潜空间中插入标签信息，强制对所学数据进行确定性约束表示（Deterministic Constrained Representation）。[ 15]
一些结构可以直接处理生成样本的质量，[ 16] [ 17] 表征学习 的效果。[ 18] [ 19]
一些结构将VAE和生成对抗网络 混合起来，以获得混合模型。[ 20] [ 21] [ 22]
Dilokthanakul, Nat; Mediano, Pedro A. M.; Garnelo, Marta; Lee, Matthew C. H.; Salimbeni, Hugh; Arulkumaran, Kai; Shanahan, Murray. Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders. 2017-01-13. arXiv:1611.02648 cs.LG ]. Kingma, Diederik P.; Rezende, Danilo J.; Mohamed, Shakir; Welling, Max. Semi-Supervised Learning with Deep Generative Models. 2014-10-31. arXiv:1406.5298 cs.LG ]. Burgess, Christopher P.; Higgins, Irina; Pal, Arka; Matthey, Loic; Watters, Nick; Desjardins, Guillaume; Lerchner, Alexander. Understanding disentangling in β-VAE. 2018-04-10. arXiv:1804.03599 stat.ML ]. Dorta, Garoe; Vicente, Sara; Agapito, Lourdes; Campbell, Neill D. F.; Simpson, Ivor. Training VAEs Under Structured Residuals. 2018-07-31. arXiv:1804.01050 stat.ML ]. Tomczak, Jakub; Welling, Max. VAE with a VampPrior . International Conference on Artificial Intelligence and Statistics (PMLR). 2018-03-31: 1214–1223 [2023-02-24 ] . arXiv:1705.07120 存档 于2021-06-24） （英语） . Razavi, Ali; Oord, Aaron van den; Vinyals, Oriol. Generating Diverse High-Fidelity Images with VQ-VAE-2. 2019-06-02. arXiv:1906.00446 cs.LG ]. Bao, Jianmin; Chen, Dong; Wen, Fang; Li, Houqiang; Hua, Gang. CVAE-GAN: Fine-Grained Image Generation Through Asymmetric Training. 2017. arXiv:1703.10155 cs.CV ]. 
![{\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right]\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/12961b671abd2f6d9f8333b9fd2c69f5729452e6)
![{\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f3930aaedee5df702f84e1571372c645eefa6572)
![{\displaystyle L_{\theta ,\phi }(x)=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln p_{\theta }(x|z)\right]-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }(\cdot ))}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e4ab2e155d237ffd569ef918817953a3ef82612c)
![{\displaystyle L_{\theta ,\phi }(x)=-{\frac {1}{2}}\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\|x-D_{\theta }(z)\|_{2}^{2}\right]-{\frac {1}{2}}\left(N\sigma _{\phi }(x)^{2}+\|E_{\phi }(x)\|_{2}^{2}-2N\ln \sigma _{\phi }(x)\right)+Const}](http://wikimedia.org/api/rest_v1/media/math/render/svg/166eb4fc2e504a10271e5bad8ba9fd0f69bc6de5)
![{\displaystyle \nabla _{\theta }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\nabla _{\theta }\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/95305894e3cfbd10c985a9569091220891523aef)
![{\displaystyle \nabla _{\phi }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bd826d47c8a9e4ac5fc59cef9026f401e9806df6)
![{\displaystyle \nabla _{\phi }\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\mathbb {E} _{\epsilon }\left[\nabla _{\phi }\ln {\frac {p_{\theta }(x,\mu _{\phi }(x)+L_{\phi }(x)\epsilon )}{q_{\phi }(\mu _{\phi }(x)+L_{\phi }(x)\epsilon |x)}}\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ab3c8e4238659d4273a37de83c8c40ce58c789fb)
