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Title: Optimal transport maps for distribution preserving operations on latent spaces of Generative Models

Paper content:
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.
Generative models such as Variational Autoencoders (VAEs) BID6 and Generative Adversarial Networks (GANs) BID3 have emerged as popular techniques for unsupervised learning of intractable distributions.
In the framework of Generative Adversarial Networks (GANs) BID3 , the generative model is obtained by jointly training a generator G and a discriminator D in an adversarial manner.
The discriminator is trained to classify synthetic samples from real ones, whereas the generator is trained to map samples drawn from a fixed prior distribution to synthetic examples which fool the discriminator.
Variational Autoencoders (VAEs) BID6 are also trained for a fixed prior distribution, but this is done through the loss of an Autoencoder that minimizes the variational lower bound of the data likelihood.
For both VAEs and GANs, using some data X we end up with a trained generator G, that is supposed to map latent samples z from the fixed prior distribution to output samples G(z) which (hopefully) have the same distribution as the data.In order to understand and visualize the learned model G(z), it is a common practice in the literature of generative models to explore how the output G(z) behaves under various arithmetic operations on the latent samples z.
In this paper, we show that the operations typically used so far, such as linear interpolation BID3 , spherical interpolation (White, 2016) , vicinity sampling and vector arithmetic BID12 , cause a distribution mismatch between the latent prior distribution and the results of the operations.
This is problematic, since the generator G was trained on a fixed prior and expects to see inputs with statistics consistent with that distribution.
We show that this, somewhat paradoxically, is also a problem if the support of resulting (mismatched) distribution is within the support of a uniformly distributed prior, whose points all have equal likelihood during training.To address this, we propose to use distribution matching transport maps, to obtain analogous latent space operations (e.g. interpolation, vicinity sampling) which preserve the prior distribution of the latent space, while minimally changing the original operation.
In Figure 1 we showcase how our proposed technique gives an interpolation operator which avoids distribution mismatch when interpolating between samples of a uniform distribution.
The points of the (red) matched trajectories samples from prior linear matched (ours) spherical (a) Uniform prior: Trajectories of linear interpolation, our matched interpolation and the spherical interpolation (White, 2016) .
(White, 2016)
Figure 1: We show examples of distribution mismatches induced by the previous interpolation schemes when using a uniform prior in two dimensions.
Our matched interpolation avoids this with a minimal modification to the linear trajectory, traversing through the space such that all points along the path are distributed identically to the prior.are obtained as minimal deviations (in expectation of l 1 distance) from the the points of the (blue) linear trajectory.
We have shown that the common latent space operations used for Generative Models induce distribution mismatch from the prior distribution the models were trained for.
This problem has been mostly ignored by the literature so far, partially due to the belief that this should not be a problem for uniform priors.
However, our statistical and experimental analysis shows that the problem is real, with the operations used so far producing significantly lower quality samples compared to their inputs.
To address the distribution mismatch, we propose to use optimal transport to minimally modify (in l 1 distance) the operations such that they fully preserve the prior distribution.
We give analytical formulas of the resulting (matched) operations for various examples, which are easily implemented.
The matched operators give a significantly higher quality samples compared to the originals, having the potential to become standard tools for evaluating and exploring generative models.
We note that the analysis here can bee seen as a more rigorous version of an observation made by White (2016) , who experimentally show that there is a significant difference between the average norm of the midpoint of linear interpolation and the points of the prior, for uniform and Gaussian distributions.Suppose our latent space has a prior with DISPLAYFORM0 In this case, we can look at the squared norm DISPLAYFORM1 From the Central Limit Theorem (CLT), we know that as d → ∞, DISPLAYFORM2 in distribution.
Thus, assuming d is large enough such that we are close to convergence, we can approximate the distribution of z 2 as N (dµ Z 2 , dσ 2 Z 2 ).
In particular, this implies that almost all points lie on a relatively thin spherical shell, since the mean grows as O(d) whereas the standard deviation grows only as O( DISPLAYFORM3 We note that this property is well known for i.i.d Gaussian entries (see e.g. Ex. 6.14 in MacKay FORMULA5 ).
For Uniform distribution on the hypercube it is also well known that the mass is concentrated in the corner points (which is consistent with the claim here since the corner points lie on a sphere).Now
consider an operator such as the midpoint of linear interpolation, y = DISPLAYFORM4 In this case, we can compute: DISPLAYFORM5 Thus, the distribution of y 2 can be approximated with N ( DISPLAYFORM6 . Therefore
, y also mostly lies on a spherical shell, but with a different radius than z. In fact,
the shells will intersect at regions which have a vanishing probability for large d. In other
words, when looking at the squared norm y 2 , y 2 is a (strong) outlier with respect to the distribution of z 2 .

assistant: text: Operations in the GAN latent space can induce a distribution mismatch compared to the training distribution, and we address this using optimal transport to match the distributions.