OPTIMAL TRANSPORT MAPS FOR DISTRIBUTION PRESERVING OPERATIONS ON LATENT SPACES OF GENERATIVE MODELS
Year: 2019

Abstract:
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. However, the latent space operations commonly used in the literature so far induce a distribution mismatch between the resulting outputs and the prior distribution the model was trained on. Previous works have attempted to reduce this mismatch with heuristic modification to the operations or by changing the latent distribution and re-training models. In this paper, we propose a framework for modifying the latent space operations such that the distribution mismatch is fully eliminated. Our approach is based on optimal transport maps, which adapt the latent space operations such that they fully match the prior distribution, while minimally modifying the original operation. Our matched operations are readily obtained for the commonly used operations and distributions and require no adjustment to the training procedure.

heading: 1 INTRODUCTION & RELATED WORK
text: Generative models such as Variational Autoencoders (VAEs) (Kingma & Welling, 2013) and Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have emerged as popular techniques for unsupervised learning of intractable distributions. In the framework of Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), 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) (Kingma & Welling, 2013) 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. However, the operations typically used so far, such as linear interpolation (Goodfellow et al., 2014), spherical interpolation (White, 2016), vicinity sampling and vector arithmetic (Radford et al., 2015), 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.
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 are obtained as minimal deviations (in expectation of l1 distance) from the the points of the (blue) linear trajectory.

heading: 1.1 DISTRIBUTION MISMATCH AND RELATED APPROACHES
text: In the literature there are dozens of papers that use sample operations to explore the learned models (Bengio et al. (2013); Goodfellow et al. (2014); Dosovitskiy et al. (2015); Reed et al. (2016); Brock et al. (2016); Reed et al. (2016) to name a few), but most of them have ignored the problem of distribution mismatch. Kingma & Welling (2013) and Makhzani et al. (2015) sidestep the problem when visualizing their models, by not performing operations on latent samples, but instead restrict the latent space to 2-d and uniformly sample the percentiles of the distribution on a 2-d grid. This way, the samples have statistics that are consistent with the prior distribution. However, this approach does not scale up to higher dimensions - whereas the latent spaces used in the literature can have hundreds of dimensions.
White (2016) experimentally observe that there is a distribution mismatch between the norm for points drawn from uniform or Gaussian distribution and points obtained with linear interpolation (SLERP), and (heuristically) propose to use a so-called spherical linear interpolation to reduce the mismatch, obtaining higher quality interpolated samples.
While SLERP has been subjectively observed to produce better looking samples than linear interpolation and is now commonly, its heuristic nature has limited it from fully replacing the linear interpolation. Furthermore, while perhaps possible it is not obvious how to generalize it to other operations, such as vicinity sampling, n-point interpolation and random walk. In Section 2 we show that for interpolation, in high dimensions SLERP tends to approximately perform distribution matching the approach taken by our framework which can explain why it works well in practice.
Kilcher et al. (2018) further analyze the (norm) distribution mismatch observed by White (2016) (in terms of KL-Divergence) for the special case of Gaussian priors, and propose an alternative prior distribution with dependent components which produces less (but still nonzero) distribution mismatch for linear interpolation, at the cost of needing to re-train and re-tune the generative models.
In contrast, we propose a framework which allows one to adapt generic operations, such that they fully preserve the original prior distribution while being faithful to the original operation. Thus the KL-Divergence between the prior and the distribution of the results from our operations is zero.
The approach works as follows: we are given a ‘desired’ operation, such as linear interpolation y = tz1 + (1− t)z2, t ∈ [0, 1]. Since the distribution of y does not match the prior distribution of z, we search for a warping f : Rd → Rd, such that ỹ = f(y) has the same distribution as z. In order to have the modification ỹ as faithful as possible to the original operation y, we use optimal transform
Published as a conference paper at ICLR 2019 Operation Expression 2-point interpolation y = tz1 + (1− t)z2 , t ∈ [0, 1] n-point interpolation y = ∑n i=1 tizi with ∑ i ti = 1
maps (Santambrogio, 2015; Villani, 2003; 2008) to find a minimal modification of y which recovers the prior distribution z.
This is illustrated in Figure 1a, where each point ỹ of the matched curve is obtained by warping a corresponding point y of the linear trajectory, while not deviating too far from the line.

heading: 2 FROM DISTRIBUTION MISMATCH TO OPTIMAL TRANSPORT
text: With implicit models such as GANs (Goodfellow et al., 2014) and VAEs (Kingma & Welling, 2013), we use the data X , drawn from an unknown random variable x, to learn a generator G : Rd 7→ Rd′ with respect to a fixed prior distribution pz , such that G(z) approximates x. Once the model is trained, we can sample from it by feeding latent samples z through G.
We now bring our attention to operations on latent samples z1, · · · , zk from pz , i.e. mappings
κ : Rd × · · · × Rd → Rd. (1)
We give a few examples of such operations in Table 1.
Since the inputs to the operations are random variables, their output y = κ(z1, · · · , zk) is also a random variable (commonly referred to as a statistic). While we typically perform these operations on realized (i.e. observed) samples, our analysis is done through the underlying random variable y. The same treatment is typically used to analyze other statistics over random variables, such as the sample mean, sample variance and test statistics.
In Table 1 we show example operations which have been commonly used in the literature. As discussed in the Introduction, such operations can provide valuable insight into how the trained generator G changes as one creates related samples y from some source samples. The most common such operation is the linear interpolation, which we can view as an operation
yt = tz1 + (1− t)z2, (2)
where z1, z2 are latent samples from the prior pz and yt is parameterized by t ∈ [0, 1]. Now, assume z1 and z2 are i.i.d, and let Z1, Z2 be their (scalar) first components with distribution pZ . Then the first component of yt is Yt = tZ1 + (1− t)Z2, and we can compute:
Var[Yt] = Var[tZ1 + (1− t)Z2] = t2Var[Z1] + (1− t)2Var[Z2] = (1 + 2t(t− 1))Var[Z]. (3)
Since (1+ 2t(t− 1)) 6= 1 for all t ∈ [0, 1] \ {0, 1}, it is in general impossible for yt to have the same distribution as z, which means that distribution mismatch is inevitable when using linear interpolation. A similar analysis reveals the same for all of the operations in Table 1.
This leaves us with a dilemma: we have various intuitive operations (see Table 1) which we would want to be able to perform on samples, but their resulting distribution pyt is inconsistent with the distribution pz we trained G for.
Due to the curse of dimensionality, as empirically observed by White (2016), this mismatch can be significant in high dimensions. We illustrate this in Figure 2, where we plot the distribution of the squared norm ‖yt‖2 for the midpoint t = 1/2 of linear interpolation, compared to the prior distribution ‖z‖2. With d = 100 (a typical dimensionality for the latent space), the distributions are dramatically different, having almost no common support. Kilcher et al. (2018) quantify this mismatch for Gaussian priors in terms of KL-Divergence, and show that it grows linearly with the dimension d. In Appendix A (see Supplement) we expand this analysis and show that this happens for all prior distributions with i.i.d. entries (i.e. not only Gaussian), both in terms of geometry and KL-Divergence.

heading: 2.1 DISTRIBUTION MATCHING WITH OPTIMAL TRANSPORT
text: In order to address the distribution mismatch, we propose a simple and intuitive framework for constructing distribution preserving operators, via optimal transport:
Strategy 1 (Optimal Transport Matched Operations). 1. We construct an ’intuitive’ operator y = κ(z1, · · · , zk).
2. We analytically (or numerically) compute the resulting (mismatched) distribution py
3. We search for a minimal modification ỹ = f(y) (in the sense that Ey[c(ỹ,y)] is minimal with respect to a cost c), such that distribution is brought back to the prior, i.e. pỹ = pz .
The cost function in step 3 could e.g. be the euclidean distance c(x, y) = ‖x − y‖, and is used to measure how faithful the modified operator, ỹ = f(κ(z1, · · · , zk)) is to the original operator k. Finding the map f which gives a minimal modification can be challenging, but fortunately it is a well studied problem from optimal transport theory. We refer to the modified operation ỹ as the matched version of y, with respect to the cost c and prior distribution pz .
For completeness, we introduce the key concepts of optimal transport theory in a simplified setting, i.e. assuming probability distributions are in euclidean space and skipping measure theoretical formalism. We refer to Villani (2003; 2008) and Santambrogio (2015) for a thorough and formal treatment of optimal transport.
The problem of step (3) above was first posed by Monge (1781) and can more formally be stated as: Problem 1 (Santambrogio (2015) Problem 1.1). Given probability distributions px, py , with domains X ,Y respectively, and a cost function c : X × Y → R+, we want to minimize
inf { Ex∼px [c(x, f(x))] ∣∣∣f : X → Y, f(x) ∼ py} (MP) We refer to the minimizer f∗X → Y of (MP) (if it exists), as the optimal transport map from px to py with respect to the cost c.
However, the problem remained unsolved until a relaxed problem was studied by Kantorovich (1942): Problem 2 (Santambrogio (2015) Problem 1.2). Given probability distributions px, py , with domains X ,Y respectively, and a cost function c : X × Y → R+, we want to minimize
inf { E(x,y)∼px,y [c(x,y)] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} , (KP) where (x,y) ∼ px,y,x ∼ px,y ∼ py denotes that (x,y) have a joint distribution px,y which has (previously specified) marginals px and py .
We refer to the joint px,y which minimizes (KP) as the optimal transport plan from px to py with respect to the cost c.
The key difference is to relax the deterministic relationship between x and f(x) to a joint probability distribution px,y with marginals px and py for x and y. In the case of Problem 1, the minimization
might be over the empty set since it is not guaranteed that there exists a mapping f such that f(x) ∼ y. In contrast, for Problem 2, one can always construct a joint density px,y with px and py as marginals, such as the trivial construction where x and y are independent, i.e. px,y(x, y) := px(x)py(y).
Note that given a joint density px,y(x, y) over X × Y , we can view y conditioned on x = x for a fixed x as a stochastic function f(x) from X to Y , since given a fixed x do not get a specific function value f(x) but instead a random variable f(x) that depends on x, with f(x) ∼ y|x = x with density py(y|x = x) := px,y(x,y)px(x) . In this case we have (x,f(x)) ∼ px,y , so we can view the Problem KP as a relaxation of Problem MP where f is allowed to be a stochastic mapping.
While the relaxed problem of Kantorovich (KP) is much more studied in the optimal transport literature, for our purposes of constructing operators it is desirable for the mapping f to be deterministic as in (MP) (see Appendix C for a more detailed discussion on deterministic vs stochastic operations).
To this end, we will choose the cost function c such that the two problems coincide and where we can find an analytical solution f or at least an efficient numerical solution.
In particular, we note that the operators in Table 1 are all pointwise, such that if the points zi have i.i.d. components, then the result y will also have i.i.d. components.
If we combine this with the constraint for the cost c to be additive over the components of x,y, we obtain the following simplification:
Theorem 1. Suppose px and py have i.i.d components and c over X × Y = Rd × Rd decomposes as
c(x, y) = d∑ i=1 C(x(i), y(i)). (4)
Consequently, the minimization problems (MP) and (KP) turn into d identical scalar problems for the distributions pX and pY of the components of x and y:
inf { EX∼pX [C(X,T (X))] ∣∣∣T : R→ R, T (X) ∼ pY } (MP-1-D) inf { E(X,Y )∼pX,Y [C(X,Y )]
∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY } , (KP-1-D) such that an optimal transport map T for (MP-1-D) gives an optimal transport map f for (MP) by pointwise application of T , i.e. f(x)(i) := T (x(i)), and an optimal transport plan pX,Y for (KP-1-D) gives an optimal transport plan px,y(x, y) := ∏d i=1 pX,Y (x (i), y(i)) for (KP).
Proof. See Appendix.
Fortunately, under some mild constraints, the scalar problems have a known solution:
Theorem 2 (Theorem 2.9 in Santambrogio (2015)). Let h : R → R+ be convex and suppose the cost C takes the form C(x, y) = h(x− y). Given an continuous source distribution pX and a target distribution pY on R having a finite optimal transport cost in (KP-1-D), then
TmonX→Y (x) := F [−1] Y (FX(x)), (5)
defines an optimal transport map from pX to pY for (MP-1-D), where FX(x) := ∫ x −∞ pX(x ′)dx′ is the Cumulative Distribution Function (CDF) of X and F [−1]Y (y) := inf{t ∈ R|FY (t) ≥ y} is the pseudo-inverse of FY . Furthermore, the joint distribution of (X,TmonX→Y (X)) defines an optimal transport plan for (KP-1-D).
The mapping TmonX→Y (x) in Theorem 2 is non-decreasing and is known as the monotone transport map from X to Y . It is easy to verify that TmonX→Y (X) has the distribution of Y , in particular FX(X) ∼ Uniform(0, 1) and if U ∼ Uniform(0, 1) then F [−1]Y (U) ∼ Y . Now, combining Theorems 1 and 2, we obtain a concrete realization of the Strategy 1 outlined above. We choose the cost c such that it admits to Theorem 1, such as c(x,y) := ‖x − y‖1, and use an operation that is pointwise, so we just need to compute the monotone transport map in (5). That is, if
z has i.i.d components with distribution pZ , we just need to compute the component distribution pY of the result y of the operation, the CDFs FZ , FY and obtain
TmonY→Z(y) := F [−1] Z (FY (y)) (6)
as the component-wise modification of y, i.e. ỹ(i) := TmonY→Z(y (i)).
In Figure 3 we show the monotone transport map for the linear interpolation y = tz1 + (1− t)z2 for various values of t. The detailed calculations and examples for various operations are given in Appendix B, for both Uniform and Gaussian priors.

heading: 3 SIMULATIONS
text: To validate the correctness of the matched operators computed in Appendix B, we numerically simulate the distributions for toy examples, as well as prior distributions typically used in the literature.
Priors vs. interpolations in 2-D For Figure 1, we sample 1 million pairs of points in two dimension, from a uniform prior (on [−1, 1]2), and estimate numerically the midpoint distribution of linear interpolation, our proposed matched interpolation and the spherical interpolation of White (2016). It is reassuring to see that the matched interpolation gives midpoints which are identically distributed to the prior. In contrast, the linear interpolation condenses more towards the origin, forming a pyramidshaped distribution (the result of convolving two boxes in 2-d). Since the spherical interpolation of White (2016) follows a great circle with varying radius between the two points, we see that the resulting distribution has a “hole” in it, “circling” around the origin for both priors.
Priors vs. interpolations in 100-D For Figure 2, we sample 1 million pairs of points in d = 100 dimensions, using either i.i.d. uniform components on [−1, 1] or Gaussian N (0, 1) and compute the distribution of the squared norm of the midpoints. We see there is a dramatic difference between vector lengths in the prior and the midpoints of linear interpolation, with only minimal overlap. We also see that the spherical interpolation (SLERP) is approximately matching the prior (norm) distribution, having a matching first moment, but otherwise also induces a distribution mismatch. In contrast, our matched interpolation, fully preserves the prior distribution and perfectly aligns. We note that this setting (d = 100, uniform or Gaussian) is commonly used in the literature.

heading: 4 EXPERIMENTS
text: Setup We used DCGAN (Radford et al., 2015) generative models trained on LSUN bedrooms (Yu et al., 2015), CelebA (Liu et al., 2015) and LLD (Sage et al., 2017; 2018), an icon dataset, to qualitatively evaluate. For LSUN, the model was trained for two different output resolutions, providing 64 × 64 pixel and a 128 × 128 pixel output images (where the latter is used in figures containing larger sample images). The models for LSUN and the icon dataset where both trained on a uniform latent prior distribution, while for CelebA a Gaussian prior was used. The dimensionality of the latent space is 100 for both LSUN and CelebA, and 512 for the model trained on the icon model. Furthermore we use improved Wasserstein GAN (iWGAN) with gradient penalty (Gulrajani et al.,
Published as a conference paper at ICLR 2019 Dataset CIFAR-10 LLD-icon LSUN CelebA Model iWGAN DCGAN DCGAN DCGAN Prior Gaussian, 128-D Uniform, 100-D Uniform, 100-D Gaussian, 100-D
Inception scores for midpoints: random samples 7.90± 0.11 3.70± 0.09 3.90± 0.08 2.05± 0.04 2-point linear 7.12± 0.08 (-10%) 3.56± 0.06 (-4%) 3.57± 0.07 (-8%) 1.71± 0.02 (-17%) 2-point matched 7.89± 0.08 3.69± 0.08 3.89± 0.08 2.04± 0.03 4-point linear 5.84± 0.08 (-26%) 3.45± 0.08 (-7%) 2.95± 0.06 (-24%) 1.46± 0.01 (-29%) 4-point matched 7.91± 0.09 3.69± 0.10 3.91± 0.10 2.04± 0.04
Table 2: Inception scores on LLD-icon, LSUN, CIFAR-10 and CelebA for the midpoints of linear interpolation and its matched counterpart. Scores are reported as mean ± standard deviation (relative change in %). Our matched variants fully recover from the (up to 29%) score drop of the linear interpolation, giving the same quality as random samples.
Prior Perturbation ‖ · ‖2 Perturbation ‖ · ‖1 Gaussian, 100-D 0.2463 0.2460 Uniform, 100-D 0.2377 0.2477 Gaussian, 128-D 0.2470 0.2460 Uniform, 128-D 0.2384 0.2479
Table 3: We measure over the average (normalized) perturbation ‖ỹ − y‖p/‖y‖p incurred by our matched interpolation for the latent spaces used in Table 2, for p = 1, 2.
2017) trained on CIFAR-10 at 32 × 32 pixels with a 128-dimensional Gaussian prior to compute inception scores.

heading: 4.1 QUANTITATIVE RESULTS
text: To measure the effect of the distribution mismatch, we quantitatively evaluate using the Inception score(Salimans et al., 2016). In Table 2 we compare the Inception score of our trained models (i.e. using random samples from the prior) with the score when sampling midpoints from the 2-point and 4-point interpolations described above, reporting mean and standard deviation with 50,000 samples, as well as relative change to the original model scores if they are significant. Compared to the original scores of the trained models (random samples), our matched operations are statistically indistinguishable (as expected) while the linear interpolation gives a significantly lower score in all settings (up to 29% lower).
However, this is not surprising, since our matched operations are guaranteed to produce samples that come from the same distribution as the random samples.
To quantify the effect our matching procedure has on the original operation, in Table 3 we compute the perturbation incurred when warping the linear interpolation y to the matched counterpart ỹ for 2-point interpolation on the latent spaces used in Table 2. We compute the normalized perturbation ‖ỹt − yt‖p/‖yt‖p (with p = 1 corresponding to l1 distance and p = 2 to l2 distance), over N = 100000 interpolation points yt = tz1 + (1− t)z2 where z1, z2 are sampled from the prior and t ∈ [0, 1] sampled uniformly. We observe that for all priors and both metrics, the perturbation is in the range 0.23− 0.25, i.e. less than a one fourth of ‖yt‖.

heading: 4.2 QUALITATIVE RESULTS
text: In the following, we will qualitatively show that our matched operations behave as expected, and that there is a visual difference between the original operations and the matched counterparts. To this end, the generator output for latent samples produced with linear interpolation, SLERP (spherical linear interpolation) of White (2016) and our proposed matched interpolation will be compared.
2-point interpolation We begin with the classic example of 2-point interpolation: Figure 4 shows three examples per dataset for an interpolation between 2 points in latent space. Each example is first done via linear interpolation, then SLERP and finally matched interpolation. It is immediately obvious in Figures 4a and 4b that linear interpolation produces inferior results with generally more blurry, less saturated and less detailed output images.
The SLERP heuristic and matched interpolation are slightly different visually, but we do not observe a difference in visual quality. However, we stress that the goal of this work is to construct operations in a principled manner, whose samples are consistent with the generative model. In the case of linear
interpolation (our framework generalizes to more operations, see below and Appendix), the SLERP heuristic tends to work well in practice but we provide a principled alternative.
4-point interpolation An even stronger effect can be observed when we do 4-point interpolation, showcased in Figure 5 (LSUN) and Figure 8 (LLD icons). The higher resolution of the LSUN output highlights the very apparent loss of detail and increasing prevalence of artifacts towards the midpoint in the linear version, compared to SLERP and our matched interpolation.
Midpoints (Appendix) In all cases, the point where the interpolation methods diverge the most, is at the midpoint of the interpolation where t = 0.5. Thus we provide 25 such interpolation midpoints in Figures 11 (LLD icons) and 12 (LSUN) in the Appendix for direct comparison.
Vicinity sampling (Appendix) Furthermore we provide two examples for vicinity sampling in Figures 9 and 10 in the Appendix. Analogous to the previous observations, the output under a linear operator lacks definition, sharpness and saturation when compared to both spherical and matched operators.
Random walk An interesting property of our matched vicinity sampling is that we can obtain a random walk in the latent space by applying it repeatedly: we start at a point y0 = z drawn from the prior, and then obtain point yi by sampling a single point in the vicinity of yi−1, using some fixed ’step size’ . We show an example of such a walk in Figure 6, using = 0.5. As a result of the repeated application of the vicinity sampling operation, the divergence from the prior distribution in the non-matched case becomes stronger with each step, resulting in completely unrecognizable output images on the LSUN and LLD icon models.

heading: 5 CONCLUSIONS
text: We proposed a framework that fully eliminates the distribution mismatch in the common latent space operations used for generative models. Our approach uses optimal transport to minimally modify (in l1 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.

heading: ACKNOWLEDGMENTS
text: This work was partly supported by ETH Zurich General Fund (OK) and Nvidia through a hardware grant.

heading: A ON THE CURSE OF DIMENSIONALITY AND GEOMETRIC OUTLIERS
text: 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 z = [Z1, · · · , Zd] ∈ [−1, 1]d with i.i.d entries Zi ∼ Z. In this case, we can look at the squared norm
‖z‖2 = d∑ i=1 Z2i . (7)
From the Central Limit Theorem (CLT), we know that as d→∞,
√ d( 1
d ‖z‖2 − µZ2)→ N (0, σ2Z2), (8)
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 (dµZ2 , dσ2Z2). 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( √ d).
We note that this property is well known for i.i.d Gaussian entries (see e.g. Ex. 6.14 in MacKay (2003)). 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 = 12z1 + 1 2z2, with components Y (i) = 12Z (i) 1 + 1 2Z (i) 2 . Furthermore, let’s assume the component distribution pZ is symmetric around 0, such that E[Z] = 0.
In this case, we can compute:
E[(Y (i))2] = Var[ 1
2 Z
(i) 1 +
1 2 Z (i) 2 ] = 1 2 Var[Z] = 1 2 µ2Z2 (9)
Var[(Y (i))2] = Var[( 1
2 Z
(i) 1 +
1 2 Z (i) 2 ) 2] = 1 4 Var[Z2] = 1 4 σ2Z2 . (10)
Thus, the distribution of ‖y‖2 can be approximated with N ( 12dµZ2 , 1 4dσ 2 Z2).
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. This can be quantified in terms of KL-Divergence:
DKL(‖z‖2, ‖y‖2) ≈ DKL(N (dµZ2 , dσ2Z2),N ( 1
2 dµZ2 ,
1 4 dσ2Z2)) (11)
= log √ dσZ2/2√ dσZ2 + dσ2Z2 + (dµZ2 − 1 2dµZ2) 2 2 14dσ 2 Z2 − 1/2 (12) = d µ2Z2
2σ2Z2 − 1/2− log 2, (13)
so DKL(‖z‖2, ‖y‖2) grows linearly with the dimensions d.
A.1 PROOF OF THEOREM 1
Proof. We will show it for the Kantorovich problem, the Monge version is similar.
Starting from (KP), we compute inf { E(x,y)∼px,y [c(x,y)] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} (14) = inf { E(x,y)∼px,y [
d∑ i=1
C(x(i),y(i))] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} (15)
= inf { d∑ i=1 E(x,y)∼px,y [C(x (i),y(i))] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} (16) ≥
d∑ i=1 inf { E(x,y)∼px,y [C(x (i),y(i))] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} (17)
= d∑ i=1 inf { E(X,Y )∼pX,Y [C(X,Y )] ∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY } (18) = d · inf { E(X,Y )∼pX,Y [C(X,Y )]
∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY } , (19) (20)
where the inequality in (17) is due to each term being minimized separately. Now let Pd(X,Y ) be the set of joints px,y with px,y(x, y) = ∏d i=1 pX,Y (x
(i), y(i)) where pX,Y has marginals pX and pY . In this case Pd(X,Y ) is a subset of all joints px,y with marginals px and py , where the pairs (x(1),y(1)), . . . , (x(d),y(d))) are constrained to be i.i.d. Starting again from (16) can compute:
inf { d∑ i=1 E(x,y)∼px,y [C(x (i),y(i))] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py}
≤ inf { d∑ i=1 E(x,y)∼px,y [C(x (i),y(i))] ∣∣∣px,y ∈ Pd(X,Y )} (21) = inf
{ d∑ i=1 E(x,y)∼px,y [C(x (i),y(i))] ∣∣∣px,y ∈ Pd(X,Y )} (22) = inf
{ d∑ i=1 E(X,Y )∼pX,Y [C(X,Y )] ∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY} (23)
= d · inf { E(X,Y )∼pX,Y [C(X,Y )] ∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY } , (24) (25)
where the inequality in (21) is due to minimizing over a smaller set.
Since the two inequalities above are in the opposite direction, equality must hold for all of the expressions above, in particular:
inf { E(x,y)∼px,y [c(x,y)] ∣∣∣(x,y) ∼ px,y,x ∼ px,y ∼ py} (26) = d · inf { E(X,Y )∼pX,Y [C(X,Y )]
∣∣∣(X,Y ) ∼ pX,Y , X ∼ pX , Y ∼ pY } (27) Thus, (KP) and (KP-1-D) equal up to a constant, and minimizing one will minimize the other. Therefore the minimization of the former can be done over pX,Y with px,y(x, y) =∏d i=1 pX,Y (x (i), y(i)).

heading: B CALCULATIONS FOR EXAMPLES
text: In the next sections, we illustrate how to compute the matched operations for a few examples, in particular for linear interpolation and vicinity sampling, using a uniform or a Gaussian prior. We picked the examples where we can analytically compute the uniform transport map, but note that it is also easy to compute F [−1]Z and (FY (y)) numerically, since one only needs to estimate CDFs in one dimension.
Since the components of all random variables in these examples are i.i.d, for such a random vector x we will implicitly write X for a scalar random variable that has the distribution of the components of x.
When computing the monotone transport map TmonX→Y , the following Lemma is helpful. Lemma 1 (Theorem 2.5 in Santambrogio (2015)). Suppose a mapping g(x) is non-decreasing and maps a continuous distribution pX to a distribution pY , i.e.
g(X) ∼ Y, (28)
then g is the monotone transport map TmonX→Y .
According to Lemma 1, an alternative way of computing TmonX→Y is to find some g that is nondecreasing and transforms pX to pY .
EXAMPLE 1:UNIFORM LINEAR INTERPOLATION
Suppose z has uniform components Z ∼ Uniform(−1, 1). In this case, pZ(z) = 1/2 for −1 < z < 1.
Now let yt = tz1 + (1 − t)z2 denote the linear interpolation between two points z1, z2, with component distribution pYt . Due to symmetry we can assume that t > 1/2, since pYt = pY1−t . We then obtain pYt as the convolution of ptZ and p(1−t)Z , i.e. pYt = ptZ ∗ p(1−t)Z . First we note that ptZ = 1/(2t) for −t < z < t and p(1−t)Z = 1/(2(1− t)) for −(1− t) < z < 1− t. We can then compute:
pYt(y) = (ptZ ∗ p(1−t)Z)(y) (29)
= 1
2(1− t)(2t)  0 if y < −1 y + 1 if − 1 < y < −t+ (1− t) 2− 2t if − t+ (1− t) < y < t− (1− t) −y + 1 if t− (1− t) < y < 1 0 if 1 < y
(30)
(31)
The CDF FYt is then obtained by computing
FYt(y) = ∫ y −∞ pYt(y ′)dy′ (32)
= 1
2(1− t)(2t)  0 if y < −1 1 2 (y + 1)(y + 1) if − 1 < y < 1− 2t 2(1− t)(y + t) if 1− 2t < y < 2t− 1 2(1− t)(3t− 1) + (− 12y 2 + y + 12 (2t− 1) 2 − (2t− 1)) if 2t− 1 < y < 1
2(1− t)(2t) if 1 < y (33)
Since pZ(z) = 1/2 for |z| < 1, we have FZ(z) = 12z + 1 2 for |z| < 1. This gives F [−1] Z (p) = 2(p− 12 ).
Now, we just compose the two mappings to obtain TmonYt→Z(y) = F [−1] Z (FYt(y)).
EXAMPLE 2: UNIFORM VICINITY SAMPLING AND RANDOM WALK
Let z again have uniform components on [−1, 1]. For vicinity sampling, we want to obtain new points z′1, ·, z′k which are close to z. We thus define
z′i := z + ui, (34)
where ui also has uniform components, such that each coordinate of z′i differs at most by from z. By identifying tZ ′i = tZ + (1− t)Ui with t = 1/(1 + ), we see that tZ ′i has identical distribution to the linear interpolation Yt in the previous example. Thus gt(Z ′i) := T mon Yt→Z(tZ ′ i) will have the distribution of Z, and by Lemma1 is then the monotone transport map from Z ′i to Z.
EXAMPLE 3: GAUSSIAN LINEAR INTERPOLATION, VICINITY SAMPLING AND ANALOGIES
Suppose z has components Z ∼ N (0, σ2). In this case, we can compute linear interpolation as before, yt = tz1+(1− t)z2. Since the sum of Gaussians is Gaussian, we get, Yt ∼ N (0, t2σ2+(1− t)2σ2). Now, it is easy to see that with a proper scaling factor, we can adjust the variance of Yt back to σ2. That is, 1√
t2+(1−t)2 Yt ∼ N (0, σ2), so by Lemma 1 gt(y) := 1√ t2+(1−t)2 y is the monotone transport
map from Yt to Z.
By adjusting the vicinity sampling operation to
z′i := z + ei, (35)
where ei ∼ N (0, 1), we can similarly find the monotone transport map g (y) = 1√1+ 2 y.
Another operation which has been used in the literature is the “analogy”, where from samples z1, z2, z3, one wants to apply the difference between z1 and z2, to z3. The transport map is then g(y) = 1√
3 y

heading: C DETERMINISTIC VS STOCHASTIC OPERATIONS
text: In Strategy 1, we only considered deterministic mappings f such that ỹ = f(y) recovers the prior distribution. However, one can also consider stochastic mappings f . One example for linear interpolation, proposed by the area chair (AC) 1, is to set ỹ|z1, z2 ∼ N (tz1 + (1 − t)z2, (1 − t2 − (1− t)2)σ2) for the case when z1, z2 ∼ N (µ, σ2) are Gaussian. This ensures that marginally ỹ ∼ N (µ, σ2), making it a valid (stochastic) modification to y that recovers the prior distribution. However, our matched interpolation has two benefits over this approach: since it is deterministic (and continuous), it means that the line between z1, z2 gets mapped to a smooth trajectory in the latent space. Furthermore, for the cost c(x,y) := ‖x− y‖1, it is optimal (see Theorem 2 ), even for the (KP) problem where f is allowed to be stochastic.
In Figure 7 we illustrate the differences between the two approaches, using a WGAN-GP model trained on CelebA with a 128 dimensional N (0, 1) latent space. We see that like our matched interpolation, the stochastic one suggested by the AC has better samples than the linear interpolation. However, due to the stochastic nature, adjacent data points do not change smoothly. In contrast our matched interpolation warps the trajectory of the linear one, which ensures that the resulting trajectory remains smooth.
1https://openreview.net/forum?id=BklCusRct7&noteId=Hkg6swibeN

heading: D ADDITIONAL EXPERIMENTS

Review summaries:
Noticing that widely used latent code interpolations for exploring the generative capabilities of VAEs and GANs have distribution mismatch problems, this paper proposes to utilize monotone transport map to exactly eliminate the distribution mismatch between modified interpolated codes and a prior distribution, assuming I.I.D. code components and a L1 code distance. More precisely, a transformation of the latent space operation is learnt with the objective that the distribution of the transformed variable match the prior distribution used in training the generative models. Optimal transport is used as a measure to minimize the two distributions. By restricting the class of cost functions used in the optimal transport formulation, the solution to the optimal transport problem (and hence the transformation function) has been shown to take a simple form (closed form in cases where cdf has a analytical form). Experiments on CIFAR-10, LLD-icon, LSUN, CelebA datasets show that, the minimally modified interpolated codes for several different interpolations produce samples with higher Inception Scores and better visual effects under an improved Wasserstein GAN than the original interpolated codes.

This paper considers the issue of distribution mismatch between the input data used for training generative models and the new data for new instance generation. Given a sample operation, the authors propose to use the so-called optimal transport to map the distribution of the new data to that of the input data that were used training. The optimal transport is essentially a monotonic transformation as the composite of the inverse of the target distribution and the source distribution.