機器學習 中，變分自編碼器（Variational Autoencoder，VAE） 是由Diederik P. Kingma和Max Welling提出的一種人工神經網絡 結構，屬於概率圖模式 和變分貝葉斯方法 。[ 1]
VAE與自編碼器 模型有關，因為兩者在結構上有一定親和力，但在目標和數學表述上有很大區別。VAE屬於概率生成模型（Probabilistic Generative Model），神經網絡僅是其中的一個組件，依照功能的不同又可分為編碼器和解碼器。編碼器可將輸入變量映射到與變分分佈的參數相對應的潛空間（Latent Space），這樣便可以產生多個遵循同一分佈的不同樣本。解碼器的功能基本相反，是從潛空間映射回輸入空間，以生成數據點。雖然噪聲模型的方差可以單獨學習而來，但它們通常都是用重參數化技巧（Reparameterization Trick）來訓練的。
此類模型最初是為無監督學習 設計的，[ 2] [ 3] 半監督學習 [ 4] [ 5] 監督學習 中也表現出卓越的有效性。[ 6]
VAE是一個分別具有先驗和噪聲分佈的生成模型，一般用最大期望算法（Expectation-Maximization meta-algorithm）來訓練。這樣可以優化數據似然的下限，用其它方法很難實現這點，且需要q分佈或變分後驗。這些q分佈通常在一個單獨的優化過程中為每個單獨數據點設定參數；而VAE則用神經網絡作為一種攤銷手段來聯合優化各個數據點，將數據點本身作為輸入，輸出變分分佈的參數。從一個已知的輸入空間映射到低維潛空間，這是一種編碼過程，因此這張神經網絡也叫「編碼器」。
解碼器則從潛空間映射回輸入空間，如作為噪聲分佈的平均值。也可以用另一個映射到方差的神經網絡，為簡單起見一般都省略掉了。這時，方差可以用梯度下降法進行優化。
優化模型常用的兩個術語是「重構誤差（reconstruction error）」和「KL散度 」。它們都來自概率模型的自由能表達式（Free Energy Expression ），因而根據噪聲分佈和數據的假定先驗而有所不同。例如，像IMAGENET這樣的標準VAE任務一般都假設具有高斯分佈噪聲，但二值化的MNIST這樣的任務則需要伯努利噪聲。自由能表達式中的KL散度使得與p分佈重疊的q分佈的概率質量最大化，但這樣可能導致出現搜尋模態（Mode-Seeking Behaviour）。自由能表達式的剩餘部分是「重構」項，需要用採樣逼近來計算其期望。[ 7]
VAE的基本框架。模型接受
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從建立概率模型的角度來看，人們希望用他們選擇的參數化概率分佈
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{\displaystyle p_{\theta }(x)=p(x|\theta )}
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{\displaystyle N(x|\mu ,\sigma )}
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邊緣化 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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{\displaystyle p_{\theta }}
聯合分佈 ，和在潛空間中的形式（也就是編碼後的
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連鎖法則 ，方程可以改寫為
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{\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz}
在香草VAE中，通常認為
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{\displaystyle p_{\theta }({x|z})}
高斯分佈 。那麼
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{\displaystyle p_{\theta }(x)}
現在，可將輸入數據和其在潛空間中的表示的映射定義為
先驗
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{\displaystyle p_{\theta }(z)}
似然值
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{\displaystyle p_{\theta }(x|z)}
後驗
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{\displaystyle p_{\theta }(z|x)}
不幸的是，對
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{\displaystyle p_{\theta }(x)}
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{\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})}
其中
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{\displaystyle q_{\phi }}
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這樣，問題就變成了找到一個好的概率自編碼器，其中條件似然分佈
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{\displaystyle p_{\theta }(x|z)}
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{\displaystyle q_{\phi }(z|x)}
下面將編碼器參數化為
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{\displaystyle E_{\phi }}
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{\displaystyle D_{\theta }}
如同每個深度學習 問題，為了通過反向傳播算法 更新神經網絡的權重，需要定義一個可微損失函數。
對於VAE，這一思想可以實現為聯合優化生成模型參數
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{\displaystyle q_{\phi }({z|x})}
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{\displaystyle p_{\theta }(z|x)}
均方誤差 和交叉熵 。
作為兩個分佈之間的距離損失，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]
剛剛定義的距離損失可擴展為
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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}}}
現在定義證據下界（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)}
給出的形式不大方便進行最大化，可以用下面的等價形式：
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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 x\sim {\mathcal {N}}(D_{\theta }(z),I)}
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{\displaystyle D_{\theta }(z)}
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{\displaystyle q_{\phi }(z|x)}
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{\displaystyle p_{\theta }(z)}
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{\displaystyle z|x\sim {\mathcal {(}}E_{\phi }(x),\sigma _{\phi }(x)^{2}I)}
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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}
重參數化技巧方案。隨機變量
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{\displaystyle {\varepsilon }}
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{\displaystyle z}
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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{\displaystyle \nabla _{\phi }}
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{\displaystyle \phi }
重參數化技巧 （也被稱為隨機反向傳播[ 10] [ 8] [ 11] [ 12]
最重要的例子是當
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{\displaystyle {\mathcal {N}}(\mu _{\phi }(x),\Sigma _{\phi }(x))}
重參數化技巧之後的VAE方案 可以通過讓
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{\displaystyle {\boldsymbol {\varepsilon }}\sim {\mathcal {N}}(0,{\boldsymbol {I}})}
隨機數生成 器」來實現重參數化，並將
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{\displaystyle z=\mu _{\phi }(x)+L_{\phi }(x)\epsilon }
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{\displaystyle L_{\phi }(x)}
科列斯基分解 得到：
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{\displaystyle \Sigma _{\phi }(x)=L_{\phi }(x)L_{\phi }(x)^{T}}
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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]}
隨機梯度下降法 了。
由於我們重參數化了
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{\displaystyle q_{\phi }(z|x)}
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{\displaystyle q_{0}}
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{\displaystyle \ln q_{\phi }(z|x)=\ln q_{0}(\epsilon )-\ln |\det(\partial _{\epsilon }z)|}
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{\displaystyle \ln q_{\phi }(z|x)=-{\frac {1}{2}}\|\epsilon \|^{2}-\ln |\det L_{\phi }(x)|-{\frac {n}{2}}\ln(2\pi )}
許多VAE的應用和擴展已被用來使其適應其他領域，並提升性能。
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{\displaystyle \beta }
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[ 13] [ 14]
條件性VAE（CVAE）在潛空間中插入標籤信息，強制對所學數據進行確定性約束表示（Deterministic Constrained Representation）。[ 15]
一些結構可以直接處理生成樣本的質量，[ 16] [ 17] 表徵學習 的效果。[ 18] [ 19]
一些結構將VAE和生成對抗網絡 混合起來，以獲得混合模型。[ 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 存檔 於2021-06-24） （英語） . 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)
