# LSGAN

Introduced by Mao et al. in Least Squares Generative Adversarial Networks

LSGAN, or Least Squares GAN, is a type of generative adversarial network that adopts the least squares loss function for the discriminator. Minimizing the objective function of LSGAN yields minimizing the Pearson $\chi^{2}$ divergence. The objective function can be defined as:

$$\min_{D}V_{LSGAN}\left(D\right) = \frac{1}{2}\mathbb{E}_{\mathbf{x} \sim p_{data}\left(\mathbf{x}\right)}\left[\left(D\left(\mathbf{x}\right) - b\right)^{2}\right] + \frac{1}{2}\mathbb{E}_{\mathbf{z}\sim p_{\mathbf{z}}\left(\mathbf{z}\right)}\left[\left(D\left(G\left(\mathbf{z}\right)\right) - a\right)^{2}\right]$$

$$\min_{G}V_{LSGAN}\left(G\right) = \frac{1}{2}\mathbb{E}_{\mathbf{z} \sim p_{\mathbf{z}}\left(\mathbf{z}\right)}\left[\left(D\left(G\left(\mathbf{z}\right)\right) - c\right)^{2}\right]$$

where $a$ and $b$ are the labels for fake data and real data and $c$ denotes the value that $G$ wants $D$ to believe for fake data.

#### Papers

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