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.
The authors demonstrate experimentally a problem with the way common latent space operations such as linear interpolation are performed for GANs and VAEs. They propose a solution based on matching distributions using optimal transport. Quite heavy machinery to solve a fairly simple problem, but their approach is practical and effective experimentally (though the gain over the simple SLERP heuristic is often marginal). The problem they describe (and so the solution) deserves to be more widely known.
Major comments:
The paper is quite verbose, probably unnecessarily so. Firstly, the authors devote over 2 pages to examples that distribution mismatches can arise in synthetic cases (section 2). This point is well made by a single example (e.g. section 2.2) and the interesting part is that this is also an issue in practice (experimental section). Secondly, the authors spend a lot of space on the precise derivation of the optimal transport map for the uniform distribution. The fact that the optimal transport computation decomposes across dimensions for pointwise operations is very relevant, and the matching of CDFs, but I think a lot of the mathematical detail could be relegated to an appendix, especially the detailed derivation of the particular CDFs.
Minor comments:
It seems worth highlighting that in practice, for the common case of a Gaussian, the proposed method for linear interpolation is just a very simple procedure that might be called "projected linear interpolation", where the generated vector is multiplied by a constant. All the optimal transport theory is nice, but it's helpful to know that this is simple to apply in practice.
Might I suggest a very simple approach to fixing the distribution mismatch issue? Train with a spherical uniform prior. When interpolating, project the linear interpolation back to the sphere. This matches distribution, and has the attractive property that the entire geodesic between two points lies in a region with typical probability density. This would also work for vicinity sampling.
In section 1, overfitting concerns seem like a strange way to motivate the desire for smoothness. Overfitting is relatively easy to compensate for, and investigating the latent space is interesting regardless.
When discussing sampling from VAEs as opposed to GANs, it would be good to mention that one has to sample from p(x | z) not just p(z).
Lots of math typos such as t - 1 should be 1 - t in (2), "V times a times r" instead of "Var" in (3) and "s times i times n" instead of "sin", etc, sqrt(1) * 2 instead of sqrt(12), inconsistent bolding of vectors. Also strange use of blackboard bold Z to mean a vector of random variables instead of the integers.
Could cite an existing source for the fact that most mass for a Gaussian is concentrated on a thin shell (section 2.2), e.g. David MacKay Information Theory, Inference and Learning Algorithms.
At the end of section 2.4, a plot of the final 1D-to-1D optimal transport function (for a few different values of t) for the uniform case would be incredibly helpful.
Section 3 should be a subsection of section 2.
For both SLERP and the proposed method, there's quite a sudden change around the midpoint of the interpolation in Figure 2. It would be interesting to plot more points around the midpoint to see the transition in more detail. (A small inkling that samples from the proposed approach might change fastest qualitatively near the midpoint of the interpolation perhaps maybe be seen in Figure 1, since the angle is changing fastest there??)