条件变分自编码器CVAE

淡淡的烟草味﹌ 2022-01-26 06:29 756阅读 0赞

转自：[https://zhuanlan.zhihu.com/p/25518643][https_zhuanlan.zhihu.com_p_25518643]

上一期探讨了[变分自编码器模型（VAEs）][VAEs]，本期继续生成模型的专题，我们来看一下条件概率版本的变分自编码器（CVAEs）。（对应的，另一类生成模型GANs也有条件概率版本，称为[CGANs][]。）

## VAE回顾 ##

VAE的目标是最大化对数似然函数

![\\sum\_i \\log p\_\{\\theta\}(\\text\{x\}^\{(i)\})=\\sum\_i KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x\}^\{(i)\})||p\_\{\\theta\}(\\text\{z\}|\\text\{x\}^\{(i)\})) + \\sum\_i \\mathcal\{L\}(\\theta,\\phi;\\text\{x\}^\{(i)\})][sum_i _log p_theta_text_x_i_sum_i KL_q_phi_text_z_text_x_i_p_theta_text_z_text_x_i_ _ _sum_i _mathcal_L_theta_phi_text_x_i]

其中，

![\\mathcal\{L\}(\\theta, \\phi; \\text\{x\}^\{(i)\}) &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x\})\} \[\\log p\_\{\\theta\}(\\text\{x,z\}) - \\log q\_\{\\phi\}(\\text\{z\}|\\text\{x\})\] \\\\ &=& -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x\}^\{(i)\})||p\_\{\\theta\}(\\text\{z\})) + \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x\}^\{(i)\})\} \\log p\_\{\\theta\}(\\text\{x\}^\{(i)\}|\\text\{z\})][mathcal_L_theta_ _phi_ _text_x_i_ _ _mathbb_E_q_phi_text_z_text_x_ _log p_theta_text_x_z_ - _log q_phi_text_z_text_x_ _ _ -KL_q_phi_text_z_text_x_i_p_theta_text_z_ _ _mathbb_E_q_phi_text_z_text_x_i_ _log p_theta_text_x_i_text_z]

由于KL散度非负，对数似然函数的变分下界即为上式中的![\\mathcal\{L\}][mathcal_L]项。一般来说，![p\_\{\\theta\}(\\text\{z\}|\\text\{x\}^\{(i)\})][p_theta_text_z_text_x_i]是未知的，或者难以获得显式表达式的，因此，直接优化对数似然函数是不可行的，一般转而优化它的变分下界，即上式中的![\\mathcal\{L\}][mathcal_L]项。Diederik P.Kingma和Max Welling提出了两个算法SGVB和AEVB去估计![\\mathcal\{L\}][mathcal_L]。

## CVAE ##

VAE用的训练集是数据![\\\{\\text\{x\}^\{(i)\}\\\}\_\{i=1\}^N][text_x_i_i_1_N]。当生成数据时，由隐变量![\\text\{z\}][text_z]控制生成数据![\\text\{x\}][text_x]，如果我们现在有的数据不只是![\\text\{x\}][text_x]，我们还有关于数据![\\text\{x\}][text_x]的一些额外信息![\\text\{y\}][text_y]，最简单的，以手写数字为例，它的标签0-9，那么我们是否能够利用上这些额外的信息呢？

**CVAE-1**

一个简答的想法，考虑条件概率分布![p\_\{\\theta\}(\\text\{x\}|\\text\{y\})][p_theta_text_x_text_y]，套用原来的VAE模型，我们不难作出以下推导：

![KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{x,y\})) &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log \\frac\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\}\{p\_\{\\theta\}(\\text\{z\}|\\text\{x,y\})\} \\\\ &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log \\frac\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\}) p\_\{\\theta\}(\\text\{x\}|\\text\{y\})\}\{p\_\{\\theta\}(\\text\{z\}|\\text\{x,y\}) p\_\{\\theta\}(\\text\{x\}|\\text\{y\})\} \\\\ &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log \\frac\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\}) p\_\{\\theta\}(\\text\{x\}|\\text\{y\})\}\{p\_\{\\theta\}(\\text\{x,z\}|\\text\{y\})\} \\\\ &=& KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\}) || p\_\{\\theta\}(\\text\{x,z\}|\\text\{y\})) + \\log p\_\{\\theta\}(\\text\{x\}|\\text\{y\}))][KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_ p_theta_text_x_text_y_p_theta_text_z_text_x_y_ p_theta_text_x_text_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_ p_theta_text_x_text_y_p_theta_text_x_z_text_y_ _ _ KL_q_phi_text_z_text_x_y_ _ p_theta_text_x_z_text_y_ _ _log p_theta_text_x_text_y]

于是

![\\log p\_\{\\theta\}(\\text\{x\}|\\text\{y\})) = KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{x,y\})) + \\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\})][log p_theta_text_x_text_y_ _ KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathcal_L_theta_phi_text_x_y]

其中，

![\\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\}) &=& -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\}) || p\_\{\\theta\}(\\text\{x,z\}|\\text\{y\})) \\\\ &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \[\\log p\_\{\\theta\}(\\text\{x,z\}|\\text\{y\}) - \\log q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\] \\\\ &=& -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{y\})) + \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log p\_\{\\theta\}(\\text\{x\}|\\text\{y,z\})][mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_ _ p_theta_text_x_z_text_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_z_text_y_ - _log q_phi_text_z_text_x_y_ _ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_y_z]

类似于VAE，套用SGVB算法，再做一下reparameterization，取适当的分布和网络，我们就得到了一个CVAE模型。

我们姑且称这个版本的CVAE为CVAE-1模型，没错，CVAE模型不止一个......

**CVAE-2**

此外，与CGAN一样，我们一般假设额外信息![\\text\{y\}][text_y]与隐变量![\\text\{z\}][text_z]没有直接的关系，因此条件概率![p\_\{\\theta\}(\\text\{z\}|\\text\{y\})=p\_\{\\theta\}(\\text\{z\})][p_theta_text_z_text_y_p_theta_text_z]，于是变分下界可以写成

![\\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\}) = -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\})) + \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log p\_\{\\theta\}(\\text\{x\}|\\text\{y,z\})][mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_y_z]

这在文献\[3\]中提到过。姑且称这个版本为CVAE-2模型。

**CVAE-3**

这就完了吗？文献\[2\]会告诉你，不要着急，我们也提出了一种CVAE。文中提出的方法不是产生数据![\\text\{x\}][text_x]，而是直接考虑预测问题：预测数据![\\text\{x\}][text_x]的标签![\\text\{y\}][text_y]。什么意思呢？它的似然函数是![p\_\{\\theta\}(\\text\{y\}|\\text\{x\})][p_theta_text_y_text_x]而不是![p\_\{\\theta\}(\\text\{x\}|\\text\{y\})][p_theta_text_x_text_y]。而这个推导也不难，事实上，把![\\text\{y\}][text_y]看成我们要生成的“数据”，![\\text\{x\}][text_x]看成是“标签”，在上面推导的结果里面直接交换![\\text\{x\}, \\text\{y\}][text_x_ _text_y]的位置，就得到了

![\\log p\_\{\\theta\}(\\text\{y\}|\\text\{x\})) = KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{x,y\})) + \\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\})][log p_theta_text_y_text_x_ _ KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathcal_L_theta_phi_text_x_y]

其中，

![\\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\}) &=& -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\}) || p\_\{\\theta\}(\\text\{y,z\}|\\text\{x\})) \\\\ &=& \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \[\\log p\_\{\\theta\}(\\text\{y,z\}|\\text\{x\}) - \\log q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\] \\\\ &=& -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{x\})) + \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log p\_\{\\theta\}(\\text\{y\}|\\text\{x,z\})][mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_ _ p_theta_text_y_z_text_x_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_y_z_text_x_ - _log q_phi_text_z_text_x_y_ _ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_y_text_x_z]

同样地，对![q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})][q_phi_text_z_text_x_y]做一下reparameterization，写成![\\text\{z\} = g\_\{\\phi\}(\\text\{x,y\},\\epsilon), \\epsilon \\sim \\mathcal\{N\}(0,I)][text_z_ _ g_phi_text_x_y_epsilon_ _epsilon _sim _mathcal_N_0_I]。再取适当的分布和网络，就可以了。值得一提的是，我们会在模型中设定适当的分布![p\_\{\\theta\}(\\text\{y\}|\\text\{x,z\})][p_theta_text_y_text_x_z]，当训练完了以后，可以把模型当成一个分类器，预测输入![\\text\{x\}][text_x]的标签：

![\\text\{y\}^\*=\\arg\\max\_\{\\text\{y\}\}p\_\{\\theta\}(\\text\{y\}|\\text\{x\},\\text\{z\}^\*), \\quad \\text\{z\}^\*=\\mathbb\{E\}\[\\text\{z\}|\\text\{x\}\]][text_y_arg_max_text_y_p_theta_text_y_text_x_text_z_ _quad _text_z_mathbb_E_text_z_text_x]

上面的预测涉及到求期望，除非有显式结果，否则一般采用均值去近似期望：

![\\text\{y\}^\*=\\arg\\max\_\{\\text\{y\}\} \\frac\{1\}\{L\} \\sum\_\{l=1\}^L p\_\{\\theta\}(\\text\{y\}|\\text\{x\},\\text\{z\}^\{(l)\}), \\quad \\text\{z\}^\{(l)\} \\sim p\_\\theta(\\text\{z\}|\\text\{x\})][text_y_arg_max_text_y_ _frac_1_L_ _sum_l_1_L p_theta_text_y_text_x_text_z_l_ _quad _text_z_l_ _sim p_theta_text_z_text_x]

姑且这个模型称为CVAE-3，它的图模型结构如下：

![v2-8a0cc590d0edc95dcd10e8adbf8772e8_hd.jpg][]

**CVAE-4**

非常抱歉地告诉你，CVAE模型还没完。文献\[3\]提出了CMMA模型（conditional multimodal autoencoder），实际上它也可以看成是条件版本的VAE。一般来说，我们考虑的CVAE或者CGAN的图模型是长这样的：

![v2-efeb6d666dbfcd69c467a643df811c66_hd.jpg][]

它的特点是![\\text\{z\}, \\text\{y\}][text_z_ _text_y]一般是相互独立的。而CMMA考虑的图模型是长这样的：

![v2-aa979cafea50272ca0017dd8d866180c_hd.jpg][]

这个模型的特点是隐变量是由额外信息![\\text\{y\}][text_y]确定的，![p\_\{\\theta\}(\\text\{x\}|\\text\{y,z\})=p\_\{\\theta\}(\\text\{x\}|\\text\{z\})][p_theta_text_x_text_y_z_p_theta_text_x_text_z]。整个推导过程跟CVAE-1一模一样，应用![p\_\{\\theta\}(\\text\{x\}|\\text\{y,z\})=p\_\{\\theta\}(\\text\{x\}|\\text\{z\})][p_theta_text_x_text_y_z_p_theta_text_x_text_z]以后，变分下界可以简化为：

![\\mathcal\{L\}(\\theta,\\phi;\\text\{x,y\}) = -KL(q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})||p\_\{\\theta\}(\\text\{z\}|\\text\{y\})) + \\mathbb\{E\}\_\{q\_\{\\phi\}(\\text\{z\}|\\text\{x,y\})\} \\log p\_\{\\theta\}(\\text\{x\}|\\text\{z\})][mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_z]

姑且称CMMA模型为CVAE-4。CVAE-4模型将标签信息编码到隐变量![\\text\{z\}][text_z]中，作者指出，这样做的效果更好。

当然，针对具体的问题，还有一些不一样的CVAE设计，例如，文献\[1\]用CVAE做半监督学习，用到的CVAE又与上面介绍的有所不同。根据具体问题，有些模型还会对目标函数添加一些惩罚项。

VAE是个贝叶斯模型，它的条件概率版本根据取条件概率的形式的不同，自然会出现多种多样的模型。

## 代码 ##

1. [RuiShu/cvae: Conditional variational autoencoder implementation in Torch][RuiShu_cvae_ Conditional variational autoencoder implementation in Torch]

2. [kastnerkyle/SciPy2015: Talk for SciPy2015 "Deep Learning: Tips From The Road"][kastnerkyle_SciPy2015_ Talk for SciPy2015 _Deep Learning_ Tips From The Road]

3. [Tutorial on Variational Autoencoders][]

4. [dpkingma/nips14-ssl: Code for reproducing results of NIPS 2014 paper "Semi-Supervised Learning with Deep Generative Models"][dpkingma_nips14-ssl_ Code for reproducing results of NIPS 2014 paper _Semi-Supervised Learning with Deep Generative Models]

5. [jramapuram/CVAE: Convolutional Variational Autoencoder][jramapuram_CVAE_ Convolutional Variational Autoencoder]

## 参考文献 ##

1. Kingma D P, Mohamed S, Rezende D J, et al. Semi-supervised learning with deep generative models\[C\]//Advances in Neural Information Processing Systems. 2014: 3581-3589.

2. Sohn K, Lee H, Yan X. Learning structured output representation using deep conditional generative models\[C\]//Advances in Neural Information Processing Systems. 2015: 3483-3491.

3. Pandey G, Dukkipati A. Variational methods for conditional multimodal learning: Generating human faces from attributes. arXiv preprint\[J\]. arXiv, 2016, 1603.

4. Walker J, Doersch C, Gupta A, et al. An uncertain future: Forecasting from static images using variational autoencoders\[C\]//European Conference on Computer Vision. Springer International Publishing, 2016: 835-851.

5. Doersch C. Tutorial on variational autoencoders\[J\]. arXiv preprint arXiv:1606.05908, 2016.

[https_zhuanlan.zhihu.com_p_25518643]: https://zhuanlan.zhihu.com/p/25518643
[VAEs]: https://zhuanlan.zhihu.com/p/25401928
[CGANs]: https://link.zhihu.com/?target=https%3A//mp.weixin.qq.com/s%3F__biz%3DMzIzOTY2NTQ5Mg%3D%3D%26mid%3D2247483957%26idx%3D1%26sn%3Dcb776f4d50f07efe1c44057bfea0fef2%26chksm%3De927ea0ede50631826d8d66f3bca656aeee314a8b374edc252c8513c70cdbf3a108700757f44%23rd
[sum_i _log p_theta_text_x_i_sum_i KL_q_phi_text_z_text_x_i_p_theta_text_z_text_x_i_ _ _sum_i _mathcal_L_theta_phi_text_x_i]: https://www.zhihu.com/equation?tex=%5Csum_i+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29%3D%5Csum_i+KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29%29+%2B+%5Csum_i+%5Cmathcal%7BL%7D%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29
[mathcal_L_theta_ _phi_ _text_x_i_ _ _mathbb_E_q_phi_text_z_text_x_ _log p_theta_text_x_z_ - _log q_phi_text_z_text_x_ _ _ -KL_q_phi_text_z_text_x_i_p_theta_text_z_ _ _mathbb_E_q_phi_text_z_text_x_i_ _log p_theta_text_x_i_text_z]: /images/20220126/17458941231e4cff9e5cd4888618de68.png%28%5Ctheta%2C+%5Cphi%3B+%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29+%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%29%7D+%5B%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%2Cz%7D%29+-+%5Clog+q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%29%5D+%5C%5C%0A%26%3D%26+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%29%29+%2B+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29%7D+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%5E%7B%28i%29%7D%7C%5Ctext%7Bz%7D%29
[mathcal_L]: /images/20220126/17458941231e4cff9e5cd4888618de68.png
[p_theta_text_z_text_x_i]: https://www.zhihu.com/equation?tex=p_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5E%7B%28i%29%7D%29
[text_x_i_i_1_N]: https://www.zhihu.com/equation?tex=%5C%7B%5Ctext%7Bx%7D%5E%7B%28i%29%7D%5C%7D_%7Bi%3D1%7D%5EN
[text_z]: /images/20220126/bb479122e47b40329d4dfd75eca9a707.png
[text_x]: /images/20220126/2ec10c7f43d54ff2914313aefcb6b68d.png
[text_y]: /images/20220126/7e6ff4b5f90c4a81ab68559650c3001e.png
[p_theta_text_x_text_y]: /images/20220126/ac4c097750424068a9f8e48ff7637c52.png
[KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_ p_theta_text_x_text_y_p_theta_text_z_text_x_y_ p_theta_text_x_text_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log _frac_q_phi_text_z_text_x_y_ p_theta_text_x_text_y_p_theta_text_x_z_text_y_ _ _ KL_q_phi_text_z_text_x_y_ _ p_theta_text_x_z_text_y_ _ _log p_theta_text_x_text_y]: https://www.zhihu.com/equation?tex=KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%29+%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+%5Cfrac%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D%7Bp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5C%5C%0A%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+%5Cfrac%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%7D%29%7D%7Bp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%7D%29%7D+%5C%5C%0A%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+%5Cfrac%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%7D%29%7D%7Bp_%7B%5Ctheta%7D%28%5Ctext%7Bx%2Cz%7D%7C%5Ctext%7By%7D%29%7D+%5C%5C%0A%26%3D%26+KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+%7C%7C+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%2Cz%7D%7C%5Ctext%7By%7D%29%29+%2B+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%7D%29%29
[log p_theta_text_x_text_y_ _ KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathcal_L_theta_phi_text_x_y]: https://www.zhihu.com/equation?tex=+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%7D%29%29+%3D+KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%29+%2B+%5Cmathcal%7BL%7D%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29
[mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_ _ p_theta_text_x_z_text_y_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_z_text_y_ - _log q_phi_text_z_text_x_y_ _ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_y_z]: /images/20220126/17458941231e4cff9e5cd4888618de68.png%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29+%26%3D%26+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+%7C%7C+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%2Cz%7D%7C%5Ctext%7By%7D%29%29+%5C%5C%0A%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5B%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%2Cz%7D%7C%5Ctext%7By%7D%29+-+%5Clog+q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%5D+%5C%5C%0A%26%3D%26+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7By%7D%29%29+%2B+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%2Cz%7D%29
[p_theta_text_z_text_y_p_theta_text_z]: /images/20220126/3013fd3e943d461caea6714871e3a9e3.png
[mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_y_z]: /images/20220126/17458941231e4cff9e5cd4888618de68.png%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29+%3D+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%29%29+%2B+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7By%2Cz%7D%29
[p_theta_text_y_text_x]: https://www.zhihu.com/equation?tex=p_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%7D%29
[text_x_ _text_y]: /images/20220126/2ec10c7f43d54ff2914313aefcb6b68d.png%2C+%5Ctext%7By%7D
[log p_theta_text_y_text_x_ _ KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_y_ _ _mathcal_L_theta_phi_text_x_y]: https://www.zhihu.com/equation?tex=+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%7D%29%29+%3D+KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%29+%2B+%5Cmathcal%7BL%7D%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29
[mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_ _ p_theta_text_y_z_text_x_ _ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_y_z_text_x_ - _log q_phi_text_z_text_x_y_ _ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_x_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_y_text_x_z]: /images/20220126/17458941231e4cff9e5cd4888618de68.png%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29+%26%3D%26+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29+%7C%7C+p_%7B%5Ctheta%7D%28%5Ctext%7By%2Cz%7D%7C%5Ctext%7Bx%7D%29%29+%5C%5C%0A%26%3D%26+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5B%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7By%2Cz%7D%7C%5Ctext%7Bx%7D%29+-+%5Clog+q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%5D+%5C%5C%0A%26%3D%26+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%29%29+%2B+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%2Cz%7D%29
[q_phi_text_z_text_x_y]: https://www.zhihu.com/equation?tex=q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29
[text_z_ _ g_phi_text_x_y_epsilon_ _epsilon _sim _mathcal_N_0_I]: /images/20220126/bb479122e47b40329d4dfd75eca9a707.png+%3D+g_%7B%5Cphi%7D%28%5Ctext%7Bx%2Cy%7D%2C%5Cepsilon%29%2C+%5Cepsilon+%5Csim+%5Cmathcal%7BN%7D%280%2CI%29
[p_theta_text_y_text_x_z]: https://www.zhihu.com/equation?tex=p_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%2Cz%7D%29
[text_y_arg_max_text_y_p_theta_text_y_text_x_text_z_ _quad _text_z_mathbb_E_text_z_text_x]: /images/20220126/7e6ff4b5f90c4a81ab68559650c3001e.png%5E%2A%3D%5Carg%5Cmax_%7B%5Ctext%7By%7D%7Dp_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%7D%2C%5Ctext%7Bz%7D%5E%2A%29%2C+%5Cquad+%5Ctext%7Bz%7D%5E%2A%3D%5Cmathbb%7BE%7D%5B%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%5D
[text_y_arg_max_text_y_ _frac_1_L_ _sum_l_1_L p_theta_text_y_text_x_text_z_l_ _quad _text_z_l_ _sim p_theta_text_z_text_x]: /images/20220126/7e6ff4b5f90c4a81ab68559650c3001e.png%5E%2A%3D%5Carg%5Cmax_%7B%5Ctext%7By%7D%7D+%5Cfrac%7B1%7D%7BL%7D+%5Csum_%7Bl%3D1%7D%5EL+p_%7B%5Ctheta%7D%28%5Ctext%7By%7D%7C%5Ctext%7Bx%7D%2C%5Ctext%7Bz%7D%5E%7B%28l%29%7D%29%2C+%5Cquad+%5Ctext%7Bz%7D%5E%7B%28l%29%7D+%5Csim+p_%5Ctheta%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%7D%29
[v2-8a0cc590d0edc95dcd10e8adbf8772e8_hd.jpg]: /images/20220126/21069c103f3649879a661ebd0871e8a8.png
[v2-efeb6d666dbfcd69c467a643df811c66_hd.jpg]: /images/20220126/ae52f9ff7c18484aa273e7c43205d050.png
[text_z_ _text_y]: /images/20220126/bb479122e47b40329d4dfd75eca9a707.png%2C+%5Ctext%7By%7D
[v2-aa979cafea50272ca0017dd8d866180c_hd.jpg]: /images/20220126/a13461d3b10940918fb3bd63442315eb.png
[p_theta_text_x_text_y_z_p_theta_text_x_text_z]: /images/20220126/e7da8d5843af4dbea65e772a1e903998.png
[mathcal_L_theta_phi_text_x_y_ _ -KL_q_phi_text_z_text_x_y_p_theta_text_z_text_y_ _ _mathbb_E_q_phi_text_z_text_x_y_ _log p_theta_text_x_text_z]: /images/20220126/17458941231e4cff9e5cd4888618de68.png%28%5Ctheta%2C%5Cphi%3B%5Ctext%7Bx%2Cy%7D%29+%3D+-KL%28q_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7C%7Cp_%7B%5Ctheta%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7By%7D%29%29+%2B+%5Cmathbb%7BE%7D_%7Bq_%7B%5Cphi%7D%28%5Ctext%7Bz%7D%7C%5Ctext%7Bx%2Cy%7D%29%7D+%5Clog+p_%7B%5Ctheta%7D%28%5Ctext%7Bx%7D%7C%5Ctext%7Bz%7D%29
[RuiShu_cvae_ Conditional variational autoencoder implementation in Torch]: https://link.zhihu.com/?target=https%3A//github.com/RuiShu/cvae
[kastnerkyle_SciPy2015_ Talk for SciPy2015 _Deep Learning_ Tips From The Road]: https://link.zhihu.com/?target=https%3A//github.com/kastnerkyle/SciPy2015
[Tutorial on Variational Autoencoders]: https://link.zhihu.com/?target=https%3A//github.com/cdoersch/vae_tutorial
[dpkingma_nips14-ssl_ Code for reproducing results of NIPS 2014 paper _Semi-Supervised Learning with Deep Generative Models]: https://link.zhihu.com/?target=https%3A//github.com/dpkingma/nips14-ssl
[jramapuram_CVAE_ Convolutional Variational Autoencoder]: https://link.zhihu.com/?target=https%3A//github.com/jramapuram/CVAE