Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks（ICCV17）

3. Formulation

2个domain记作$X$X和$Y$Y，unpaired data $\left \{ x_i \right \}_{i=1}^N, x_i\in X${xi​}i=1N​,xi​∈X，$\left \{ y_j \right \}_{j=1}^M, y_j\in Y${yj​}j=1M​,yj​∈Y

记data distribution $x\sim p_{data}(x), y\sim p_{data}(y)$x∼pdata​(x),y∼pdata​(y)

如Fig.3所示，整个model包含2个生成网络，$G:X\rightarrow Y，F:Y\rightarrow X$G:X→Y，F:Y→X，2个domain的判别器$D_X, D_Y$DX​,DY​

损失函数包括adversarial loss、cycle consistency loss

3.1. Adversarial Loss

adversarial loss定义如下
$\begin{aligned} \mathcal{L}_{GAN}\left ( G, D_Y, X, Y \right )&=\mathbb{E}_{y\sim p_{data}(y)}\left [ \log D_Y(y) \right ] \\ &+\mathbb{E}_{x\sim p_{data}(x)}\left [ \log\left ( 1-D_Y\left ( G(x) \right ) \right ) \right ] \qquad(1) \end{aligned}$LGAN​(G,DY​,X,Y)​=Ey∼pdata​(y)​[logDY​(y)]+Ex∼pdata​(x)​[log(1−DY​(G(x)))](1)​
对于$X\rightarrow Y$X→Y的生成，$\min_G\max_{D_Y}\mathcal{L}_{GAN}\left ( G, D_Y, X, Y \right )$minG​maxDY​​LGAN​(G,DY​,X,Y)
对于$Y\rightarrow X$Y→X的生成，$\min_F\max_{D_X}\mathcal{L}_{GAN}\left ( F, D_X, Y, X \right )$minF​maxDX​​LGAN​(F,DX​,Y,X)

3.2. Cycle Consistency Loss