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What is the best way to evaluate performance of Generative Adverserial Network (GAN)? Perhaps measuring the distance between two distributions or maybe something else?

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I think it depends on what exactly you're doing with the GANs. If you're generating images, the two most popular (to my knowledge) are the Inception Score [1] and Frechet Inception Distance [2]. GANs aren't my direct area of study, but I don't think you can miss starting with those two.

Actually, Goodfellow himself endorsed FID (a year ago) on twitter, so I'd start there.

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