We investigate the properties of multidimensional probability distributions in the context of latent space prior distributions of implicit generative models. Our work revolves around the phenomena arising while decoding linear interpolations between two random latent vectors -- regions of latent space in close proximity to the origin of the space are sampled causing distribution mismatch. We show that due to the Central Limit Theorem, this region is almost never sampled during the training process. As a result, linear interpolations may generate unrealistic data and their usage as a tool to check quality of the trained model is questionable. We propose to use multidimensional Cauchy distribution as the latent prior. Cauchy distribution does not satisfy the assumptions of the CLT and has a number of properties that allow it to work well in conjunction with linear interpolations. We also provide two general methods of creating non-linear interpolations that are easily applicable to a large family of common latent distributions. Finally we empirically analyze the quality of data generated from low-probability-mass regions for the DCGAN model on the CelebA dataset.
Some approaches to counteract this phenomena were proposed: recommended using spherical interpolations to avoid traversing unlikely regions; @cite_9 suggest normalizing the norms of the points along the interpolation to match the prior distribution; propose using a modified prior distribution, which saturates the origin of the latent space. gives an interesting discussion on latent space traversal using the theory of Riemannian spaces.
@cite_9
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Generative models such as Variational Auto Encoders (VAEs) and Generative Adversarial Networks (GANs) are typically trained for a fixed prior distribution in the latent space, such as uniform or Gaussian. After a trained model is obtained, one can sample the Generator in various forms for exploration and understanding, such as interpolating between two samples, sampling in the vicinity of a sample or exploring differences between a pair of samples applied to a third sample. In this paper, we show that the latent space operations used in the literature so far induce a distribution mismatch between the resulting outputs and the prior distribution the model was trained on. To address this, we propose to use distribution matching transport maps to ensure that such latent space operations preserve the prior distribution, while minimally modifying the original operation. Our experimental results validate that the proposed operations give higher quality samples compared to the original operations.