Sanjeev Arora, Yi Zhang

https://arxiv.org/abs/1706.08224

Problem

- GANs do not provide a measure of distributional fit/perplexity
- Current tests on GANS rule out the possibility that GANs are copying training, but do not indicate if it has learned the target distribution
- Support size: Loosely, the number of features that can specify a distribution - e.g. number/categories of facial features for a set of facial images. What is the support size of a distribution? In a target distribution of small support size, is it possible for a GAN to reach low training error but still not learn the support of the target distribution?

Uses a birthday paradox test. When samples are chosen from a set of size

*N*, the probability that a duplicate exists is reasonably high (close to 0.5) when the number of samples chosen

*> squareroot(N)*. This can be used to test a GAN. If the training distribution has support size

*N*, then generate samples of size

*S*. If

*S*has a reasonable number of duplicates then support size of generated distribution must be

*S^2*. If

*S^2*is close to

*N*, then the GAN is approximating the training distribution. This test fails if training distribution is highly skewed.

Results

- CIFAR-10: The support size is the vector embedding of the final layer of a CNN trained for classification. Duplicates are identified using a Euclidian similarity measure between the vector embeddings. The GAN used GAN was a stacked GAN
- Compares two measures:
- Diversity: Square of the number of samples need to detect a duplicate with probability> 0.5
- Discriminator size: Size of embedding vector in final layer of discriminator CNN.
- If a GAN is learning the distribution then diversity should be close to discriminator size.
- Results on CIFAR indicate that
- The GAN is learning lower support size than needed to represent the target distribution, however it is not copying training images
- The size of the generated distribution’s support (diversity) scales near-linearly with discriminator capacity

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