Wasserstein Auto-Encoders

11/05/2017 ∙ by Ilya Tolstikhin, et al. ∙ 0

We propose the Wasserstein Auto-Encoder (WAE)---a new algorithm for building a generative model of the data distribution. WAE minimizes a penalized form of the Wasserstein distance between the model distribution and the target distribution, which leads to a different regularizer than the one used by the Variational Auto-Encoder (VAE). This regularizer encourages the encoded training distribution to match the prior. We compare our algorithm with several other techniques and show that it is a generalization of adversarial auto-encoders (AAE). Our experiments show that WAE shares many of the properties of VAEs (stable training, encoder-decoder architecture, nice latent manifold structure) while generating samples of better quality, as measured by the FID score.

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1 Introduction

The field of representation learning was initially driven by supervised approaches, with impressive results using large labelled datasets. Unsupervised generative modeling, in contrast, used to be a domain governed by probabilistic approaches focusing on low-dimensional data. Recent years have seen a convergence of those two approaches. In the new field that formed at the intersection, variational auto-encoders (VAEs) [1] constitute one well-established approach, theoretically elegant yet with the drawback that they tend to generate blurry samples when applied to natural images. In contrast, generative adversarial networks (GANs) [3] turned out to be more impressive in terms of the visual quality of images sampled from the model, but come without an encoder, have been reported harder to train, and suffer from the “mode collapse” problem where the resulting model is unable to capture all the variability in the true data distribution. There has been a flurry of activity in assaying numerous configurations of GANs as well as combinations of VAEs and GANs. A unifying framework combining the best of GANs and VAEs in a principled way is yet to be discovered.

This work builds up on the theoretical analysis presented in [4]. Following [5, 4], we approach generative modeling from the optimal transport (OT) point of view. The OT cost [6]

is a way to measure a distance between probability distributions and provides a much weaker topology than many others, including

-divergences associated with the original GAN algorithms [7]. This is particularly important in applications, where data is usually supported on low dimensional manifolds in the input space . As a result, stronger notions of distances (such as -divergences, which capture the density ratio between distributions) often max out, providing no useful gradients for training. In contrast, OT was claimed to have a nicer behaviour [5, 8] although it requires, in its GAN-like implementation, the addition of a constraint or a regularization term into the objective.

In this work we aim at minimizing OT between the true (but unknown) data distribution and a latent variable model specified by the prior distribution of latent codes and the generative model of the data points given . Our main contributions are listed below (cf. also Figure 1):

  • A new family of regularized auto-encoders (Algorithms 1, 2 and Eq. 4), which we call Wasserstein Auto-Encoders (WAE), that minimize the optimal transport for any cost function . Similarly to VAE, the objective of WAE is composed of two terms: the -reconstruction cost and a regularizer penalizing a discrepancy between two distributions in : and a distribution of encoded data points, i.e. . When is the squared cost and is the GAN objective, WAE coincides with adversarial auto-encoders of [2].

  • Empirical evaluation of WAE on MNIST and CelebA datasets with squared cost . Our experiments show that WAE keeps the good properties of VAEs (stable training, encoder-decoder architecture, and a nice latent manifold structure) while generating samples of better quality, approaching those of GANs.

  • We propose and examine two different regularizers . One is based on GANs and adversarial training in the latent space

    . The other uses the maximum mean discrepancy, which is known to perform well when matching high-dimensional standard normal distributions

    [9]. Importantly, the second option leads to a fully adversary-free min-min optimization problem.

  • Finally, the theoretical considerations presented in [4] and used here to derive the WAE objective might be interesting in their own right. In particular, Theorem 1 shows that in the case of generative models, the primal form of is equivalent to a problem involving the optimization of a probabilistic encoder .

VAE reconstruction

(a) VAE

WAE reconstruction

(b) WAE
Figure 1: Both VAE and WAE minimize two terms: the reconstruction cost and the regularizer penalizing discrepancy between and distribution induced by the encoder . VAE forces to match for all the different input examples drawn from . This is illustrated on picture (a), where every single red ball is forced to match depicted as the white shape. Red balls start intersecting, which leads to problems with reconstruction. In contrast, WAE forces the continuous mixture to match , as depicted with the green ball in picture (b). As a result latent codes of different examples get a chance to stay far away from each other, promoting a better reconstruction.

The paper is structured as follows. In Section 2 we review a novel auto-encoder formulation for OT between and the latent variable model derived in [4]. Relaxing the resulting constrained optimization problem we arrive at an objective of Wasserstein auto-encoders. We propose two different regularizers, leading to WAE-GAN and WAE-MMD algorithms. Section 3 discusses the related work. We present the experimental results in Section 4 and conclude by pointing out some promising directions for future work.

2 Proposed method

Our new method minimizes the optimal transport cost based on the novel auto-encoder formulation (see Theorem 1 below). In the resulting optimization problem the decoder tries to accurately reconstruct the encoded training examples as measured by the cost function . The encoder tries to simultaneously achieve two conflicting goals: it tries to match the encoded distribution of training examples to the prior as measured by any specified divergence , while making sure that the latent codes provided to the decoder are informative enough to reconstruct the encoded training examples. This is schematically depicted on Fig. 1.

2.1 Preliminaries and notations

We use calligraphic letters (i.e.) for sets, capital letters (i.e.

) for random variables, and lower case letters (i.e.

) for their values. We denote probability distributions with capital letters (i.e.) and corresponding densities with lower case letters (i.e.). In this work we will consider several measures of discrepancy between probability distributions and . The class of -divergences [10] is defined by , where is any convex function satisfying . Classical examples include the Kullback-Leibler and Jensen-Shannon divergences.

2.2 Optimal transport and its dual formulations

A rich class of divergences between probability distributions is induced by the optimal transport (OT) problem [6]. Kantorovich’s formulation of the problem is given by

(1)

where is any measurable cost function and

is a set of all joint distributions of

with marginals and respectively. A particularly interesting case is when is a metric space and for . In this case , the -th root of , is called the -Wasserstein distance.

When the following Kantorovich-Rubinstein duality holds111 Note that the same symbol is used for and , but only is a number and thus the above refers to the 1-Wasserstein distance. :

(2)

where is the class of all bounded 1-Lipschitz functions on .

2.3 Application to generative models: Wasserstein auto-encoders

One way to look at modern generative models like VAEs and GANs is to postulate that they are trying to minimize certain discrepancy measures between the data distribution and the model . Unfortunately, most of the standard divergences known in the literature, including those listed above, are hard or even impossible to compute, especially when is unknown and

is parametrized by deep neural networks. Previous research provides several tricks to address this issue.

In case of minimizing the KL-divergence , or equivalently maximizing the marginal log-likelihood , the famous variational lower bound provides a theoretically grounded framework successfully employed by VAEs [1, 11]. More generally, if the goal is to minimize the -divergence (with one example being ), one can resort to its dual formulation and make use of -GANs and the adversarial training [7]. Finally, OT cost is yet another option, which can be, thanks to the celebrated Kantorovich-Rubinstein duality (2), expressed as an adversarial objective as implemented by the Wasserstein-GAN [5]. We include an extended review of all these methods in Supplementary A.

In this work we will focus on latent variable models defined by a two-step procedure, where first a code is sampled from a fixed distribution on a latent space and then is mapped to the image with a (possibly random) transformation. This results in a density of the form

(3)

assuming all involved densities are properly defined. For simplicity we will focus on non-random decoders, i.e.generative models deterministically mapping to for a given map . Similar results for random decoders can be found in Supplementary B.1.

It turns out that under this model, the OT cost takes a simpler form as the transportation plan factors through the map : instead of finding a coupling in (1) between two random variables living in the space, one distributed according to and the other one according to , it is sufficient to find a conditional distribution such that its marginal is identical to the prior distribution . This is the content of the theorem below proved in [4]. To make this paper self contained we repeat the proof in Supplementary B.

Theorem 1.

For as defined above with deterministic and any function

where is the marginal distribution of when and .

This result allows us to optimize over random encoders instead of optimizing over all couplings between and . Of course, both problems are still constrained. In order to implement a numerical solution we relax the constraints on by adding a penalty to the objective. This finally leads us to the WAE objective:

(4)

where is any nonparametric set of probabilistic encoders, is an arbitrary divergence between and , and

is a hyperparameter. Similarly to VAE, we propose to use deep neural networks to parametrize both encoders

and decoders . Note that as opposed to VAEs, the WAE formulation allows for non-random encoders deterministically mapping inputs to their latent codes.

We propose two different penalties :

GAN-based . The first option is to choose

and use the adversarial training to estimate it. Specifically, we introduce an adversary (discriminator) in the latent space

trying to separate222 We noticed that the famous “log trick” (also called “non saturating loss”) proposed by [3] leads to better results. “true” points sampled from and “fake” ones sampled from [3]. This results in the WAE-GAN described in Algorithm 1. Even though WAE-GAN falls back to the min-max problem, we move the adversary from the input (pixel) space to the latent space . On top of that, may have a nice shape with a single mode (for a Gaussian prior), in which case the task should be easier than matching an unknown, complex, and possibly multi-modal distributions as usually done in GANs. This is also a reason for our second penalty:

MMD-based . For a positive-definite reproducing kernel the following expression is called the maximum mean discrepancy (MMD):

where is the RKHS of real-valued functions mapping to . If is characteristic then defines a metric and can be used as a divergence measure. We propose to use

. Fortunately, MMD has an unbiased U-statistic estimator, which can be used in conjunction with stochastic gradient descent (SGD) methods. This results in the WAE-MMD described in Algorithm

2. It is well known that the maximum mean discrepancy performs well when matching high-dimensional standard normal distributions [9] so we expect this penalty to work especially well working with the Gaussian prior .

0:  Regularization coefficient .
   Initialize the parameters of the encoder , decoder , and latent discriminator .
  while  not converged do
      Sample from the training set Sample from the prior Sample from for Update by ascending:
Update and by descending:
  end while
Algorithm 1 Wasserstein Auto-Encoder with GAN-based penalty (WAE-GAN).
0:  Regularization coefficient , characteristic positive-definite kernel .
   Initialize the parameters of the encoder , decoder , and latent discriminator .
  while  not converged do
      Sample from the training set Sample from the prior Sample from for Update and by descending:
  end while
Algorithm 2 Wasserstein Auto-Encoder with MMD-based penalty (WAE-MMD).

We point out once again that the encoders in Algorithms 1 and 2 can be non-random, i.e.deterministically mapping input points to the latent codes. In this case for a function and in order to sample from we just need to return .

3 Related work

Literature on auto-encoders Classical unregularized auto-encoders minimize only the reconstruction cost. This results in different training points being encoded into non-overlapping zones chaotically scattered all across the space with “holes” in between where the decoder mapping has never been trained. Overall, the encoder trained in this way does not provide a useful representation and sampling from the latent space becomes hard [12].

Variational auto-encoders [1] minimize a variational bound on the KL-divergence which is composed of the reconstruction cost plus the regularizer . The regularizer captures how distinct the image by the encoder of each training example is from the prior , which is not guaranteeing that the overall encoded distribution matches like WAE does. Also, VAEs require non-degenerate (i.e.non-deterministic) Gaussian encoders and random decoders for which the term can be computed and differentiated with respect to the parameters. Later [11] proposed a way to use VAE with non-Gaussian encoders. WAE minimizes the optimal transport and allows both probabilistic and deterministic encoder-decoder pairs of any kind.

The VAE regularizer can be also equivalently written [13] as a sum of and a mutual information between the images and latent codes jointly distributed according to . This observation provides another intuitive way to explain a difference between our algorithm and VAEs: WAEs simply drop the mutual information term in the VAE regularizer.

When used with WAE-GAN is equivalent to adversarial auto-encoders (AAE) proposed by [2]. Theory of [4] (and in particular Theorem 1) thus suggests that AAEs minimize the 2-Wasserstein distance between and . This provides the first theoretical justification for AAEs known to the authors. WAE generalizes AAE in two ways: first, it can use any cost function  in the input space ; second, it can use any discrepancy measure in the latent space (for instance MMD), not necessarily the adversarial one of WAE-GAN.

Finally, [14] independently proposed a regularized auto-encoder objective similar to [4] and our (4) based on very different motivations and arguments. Following VAEs their objective (called InfoVAE) defines the reconstruction cost in the image space implicitly through the negative log likelihood term , which should be properly normalized for all . In theory VAE and InfoVAE can both induce arbitrary cost functions, however in practice this may require an estimation of the normalizing constant (partition function) which can333Two popular choices are Gaussian and Bernoulli decoders leading to pixel-wise squared and cross-entropy losses respectively. In both cases the normalizing constants can be computed in closed form and don’t depend on . be different for different values of . WAEs specify the cost explicitly and don’t constrain it in any way.

Literature on OT [15] address computing the OT cost in large scale using SGD and sampling. They approach this task either through the dual formulation, or via a regularized version of the primal. They do not discuss any implications for generative modeling. Our approach is based on the primal form of OT, we arrive at regularizers which are very different, and our main focus is on generative modeling.

The WGAN [5] minimizes the 1-Wasserstein distance for generative modeling. The authors approach this task from the dual form. Their algorithm comes without an encoder and can not be readily applied to any other cost , because the neat form of the Kantorovich-Rubinstein duality (2) holds only for . WAE approaches the same problem from the primal form, can be applied for any cost function , and comes naturally with an encoder.

In order to compute the values (1) or (2) of OT we need to handle non-trivial constraints, either on the coupling distribution or on the function being considered. Various approaches have been proposed in the literature to circumvent this difficulty. For [5] tried to implement the constraint in the dual formulation (2) by clipping the weights of the neural network . Later [8] proposed to relax the same constraint by penalizing the objective of (2) with a term which should not be greater than 1 if . In a more general OT setting of [16] proposed to penalize the objective of (1) with the KL-divergence between the coupling distribution and the product of marginals. [15] showed that this entropic regularization drops the constraints on functions in the dual formulation as opposed to (2). Finally, in the context of unbalanced optimal transport it has been proposed to relax the constraint in (1) by regularizing the objective with [17, 18], where and are marginals of . In this paper we propose to relax OT in a way similar to the unbalanced optimal transport, i.e. by adding additional divergences to the objective. However, we show that in the particular context of generative modeling, only one extra divergence is necessary.

Test interpolations          Test reconstructions               Random samples

VAE WAE-MMD WAE-GAN
Figure 2:

VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on MNIST dataset. In “test reconstructions” odd rows correspond to the real test points.

Literature on GANs Many of the GAN variations (including -GAN and WGAN) come without an encoder. Often it may be desirable to reconstruct the latent codes and use the learned manifold, in which cases these models are not applicable.

There have been many other approaches trying to blend the adversarial training of GANs with auto-encoder architectures [19, 20, 21, 22]. The approach proposed by [21] is perhaps the most relevant to our work. The authors use the discrepancy between and the distribution

of auto-encoded noise vectors as the objective for the max-min game between the encoder and decoder respectively. While the authors showed that the saddle points correspond to

, they admit that encoders and decoders trained in this way have no incentive to be reciprocal. As a workaround they propose to include an additional reconstruction term to the objective. WAE does not necessarily lead to a min-max game, uses a different penalty, and has a clear theoretical foundation.

Several works used reproducing kernels in context of GANs. [23, 24] use MMD with a fixed kernel to match and directly in the input space . These methods have been criticised to require larger mini-batches during training: estimating requires number of samples roughly proportional to the dimensionality of the input space [25] which is typically larger than . [26] take a similar approach but further train

adversarially so as to arrive at a meaningful loss function. WAE-MMD uses MMD to match

to the prior in the latent space . Typically has no more than dimensions and is Gaussian, which allows us to use regular mini-batch sizes to accurately estimate MMD.

Test interpolations          Test reconstructions               Random samples VAE WAE-MMD WAE-GAN
Figure 3: VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on CelebA dataset. In “test reconstructions” odd rows correspond to the real test points.

4 Experiments

In this section we empirically evaluate444 The code is available at github.com/tolstikhin/wae. the proposed WAE model. We would like to test if WAE can simultaneously achieve (i) accurate reconstructions of data points, (ii) reasonable geometry of the latent manifold, and (iii) random samples of good (visual) quality. Importantly, the model should generalize well: requirements (i) and (ii) should be met on both training and test data. We trained WAE-GAN and WAE-MMD (Algorithms 1 and 2) on two real-world datasets: MNIST [27] consisting of 70k images and CelebA [28] containing roughly 203k images.

Experimental setup In all reported experiments we used Euclidian latent spaces for various depending on the complexity of the dataset, isotropic Gaussian prior distributions over , and a squared cost function for data points . We used deterministic encoder-decoder pairs, Adam [29] with , and convolutional deep neural network architectures for encoder mapping and decoder mapping similar to the DCGAN ones reported by [30]

with batch normalization

[31]. We tried various values of and noticed that seems to work good across all datasets we considered.

Since we are using deterministic encoders, choosing larger than intrinsic dimensionality of the dataset would force the encoded distribution to live on a manifold in . This would make matching to impossible if is Gaussian and may lead to numerical instabilities. We use for MNIST and for CelebA which seems to work reasonably well.

We also report results of VAEs. VAEs used the same latent spaces as discussed above and standard Gaussian priors . We used Gaussian encoders with mean and diagonal covariance . For MNIST we used Bernoulli decoders parametrized by and for CelebA the Gaussian decoders with mean . Functions , , and were parametrized by deep nets of the same architectures as in WAE.

WAE-GAN and WAE-MMD specifics In WAE-GAN we used discriminator

composed of several fully connected layers with ReLu. We tried WAE-MMD with the RBF kernel but observed that it fails to penalize the outliers of

because of the quick tail decay. If the codes for some of the training points end up far away from the support of (which may happen in the early stages of training) the corresponding terms in the U-statistic will quickly approach zero and provide no gradient for those outliers. This could be avoided by choosing the kernel bandwidth

in a data-dependent manner, however in this case per-minibatch U-statistic would not provide an unbiased estimate for the gradient. Instead, we used the

inverse multiquadratics kernel which is also characteristic and has much heavier tails. In all experiments we used , which is the expected squared distance between two multivariate Gaussian vectors drawn from . This significantly improved the performance compared to the RBF kernel (even the one with ). Trained models are presented in Figures 2 and 3. Further details are presented in Supplementary C.

Random samples are generated by sampling and decoding the resulting noise vectors into . As expected, in our experiments we observed that for both WAE-GAN and WAE-MMD the quality of samples strongly depends on how accurately matches . To see this, notice that during training the decoder function is presented only with encoded versions of the data points . Indeed, the decoder is trained on samples from and thus there is no reason to expect good results when feeding it with samples from . In our experiments we noticed that even slight differences between and may affect the quality of samples.

Algorithm FID Sharpness
VAE 63
WAE-MMD 55
WAE-GAN 42
True data 2
Table 1: FID (smaller is better) and sharpness (larger is better) scores for samples of various models for CelebA.

In some cases WAE-GAN seems to lead to a better matching and generates better samples than WAE-MMD. However, due to adversarial training WAE-GAN is less stable than WAE-MMD, which has a very stable training much like VAE.

In order to quantitatively assess the quality of the generated images, we use the Fréchet Inception Distance introduced by [32] and report the results on CelebA based on

samples. We also heuristically evaluate the

sharpness of generated samples 555 Every image is converted to greyscale and convolved with the Laplace filter

, which acts as an edge detector. We compute the variance of the resulting activations and average these values across 1000 images sampled from a given model. The blurrier the image, the less edges it has, and the more activations will be close to zero, leading to smaller variances.

using the Laplace filter. The numbers, summarized in Table 1, show that WAE-MMD has samples of slightly better quality than VAE, while WAE-GAN achieves the best results overall.

Test reconstructions and interpolations. We take random points from the held out test set and report their auto-encoded versions . Next, pairs of different data points are sampled randomly from the held out test set and encoded: , . We linearly interpolate between and with equally-sized steps in the latent space and show decoded images.

5 Conclusion

Using the optimal transport cost, we have derived Wasserstein auto-encoders—a new family of algorithms for building generative models. We discussed their relations to other probabilistic modeling techniques. We conducted experiments using two particular implementations of the proposed method, showing that in comparison to VAEs, the images sampled from the trained WAE models are of better quality, without compromising the stability of training and the quality of reconstruction. Future work will include further exploration of the criteria for matching the encoded distribution to the prior distribution , assaying the possibility of adversarially training the cost function in the input space , and a theoretical analysis of the dual formulations for WAE-GAN and WAE-MMD.

Acknowledgments

The authors are thankful to Carl Johann Simon-Gabriel, Mateo Rojas-Carulla, Arthur Gretton, Paul Rubenstein, and Fei Sha for stimulating discussions.

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Appendix A Implicit generative models: a short tour of GANs and VAEs

Even though GANs and VAEs are quite different—both in terms of the conceptual frameworks and empirical performance—they share important features: (a) both can be trained by sampling from the model without knowing an analytical form of its density and (b) both can be scaled up with SGD. As a result, it becomes possible to use highly flexible implicit models defined by a two-step procedure, where first a code is sampled from a fixed distribution on a latent space and then is mapped to the image with a (possibly random) transformation . This results in latent variable models of the form (3).

These models are indeed easy to sample and, provided can be differentiated analytically with respect to its parameters, can be trained with SGD. The field is growing rapidly and numerous variations of VAEs and GANs are available in the literature. Next we introduce and compare several of them.

The original generative adversarial network (GAN) [3] approach minimizes

(5)

with respect to a deterministic decoder , where is any non-parametric class of choice. It is known that and the inequality turns into identity in the nonparametric limit, that is when the class becomes rich enough to represent all functions mapping to . Hence, GANs are minimizing a lower bound on the JS-divergence. However, GANs are not only linked to the JS-divergence: the -GAN approach [7] showed that a slight modification of the objective (5) allows to lower bound any desired -divergence in a similar way. In practice, both decoder and discriminator are trained in alternating SGD steps. Stopping criteria as well as adequate evaluation of the trained GAN models remain open questions.

Recently, the authors of [5] argued that the 1-Wasserstein distance , which is known to induce a much weaker topology than , may be better suited for generative modeling. When and are supported on largely disjoint low-dimensional manifolds (which may be the case in applications), , , and other strong distances between and max out and no longer provide useful gradients for

. This “vanishing gradient” problem necessitates complicated scheduling between the

/ updates. In contrast, is still sensible in these cases and provides stable gradients. The Wasserstein GAN (WGAN) minimizes

where is any subset of 1-Lipschitz functions on . It follows from (2) that and thus WGAN is minimizing a lower bound on the 1-Wasserstein distance.

Variational auto-encoders (VAE) [1] utilize models of the form (3) and minimize

(6)

with respect to a random decoder mapping . The conditional distribution is often parametrized by a deep net and can have any form as long as its density can be computed and differentiated with respect to the parameters of . A typical choice is to use Gaussians . If is the set of allconditional probability distributions , the objective of VAE coincides with the negative marginal log-likelihood . However, in order to make the term of (6) tractable in closed form, the original implementation of VAE uses a standard normal and restricts

to a class of Gaussian distributions

with mean and diagonal covariance  parametrized by deep nets. As a consequence, VAE is minimizing an upper bound on the negative log-likelihood or, equivalently, on the KL-divergence .

One possible way to reduce the gap between the true negative log-likelihood and the upper bound provided by is to enlarge the class . Adversarial variational Bayes (AVB) [11] follows this argument by employing the idea of GANs. Given any point , a noise , and any fixed transformation , a random variable implicitly defines one particular conditional distribution . AVB allows to contain all such distributions for different choices of , replaces the intractable term in (6) by the adversarial approximation corresponding to the KL-divergence, and proposes to minimize666 The authors of AVB [11] note that using -GAN as described above actually results in “unstable training”. Instead, following the approach of [33], they use a trained discriminator resulting from the objective (5) to approximate the ratio of densities and then directly estimate the KL divergence .

(7)

The term in (6) may be viewed as a regularizer. Indeed, VAE reduces to the classical unregularized auto-encoder if this term is dropped, minimizing the reconstruction cost of the encoder-decoder pair . This often results in different training points being encoded into non-overlapping zones chaotically scattered all across the space with “holes” in between where the decoder mapping has never been trained. Overall, the encoder trained in this way does not provide a useful representation and sampling from the latent space becomes hard [12].

Adversarial auto-encoders (AAE) [2] replace the term in (6) with another regularizer:

(8)

where is the marginal distribution of when first is sampled from and then is sampled from , also known as the aggregated posterior [2]. Similarly to AVB, there is no clear link to log-likelihood, as . The authors of [2] argue that matching to in this way ensures that there are no “holes” left in the latent space and generates reasonable samples whenever . They also report an equally good performance of different types of conditional distributions , including Gaussians as used in VAEs, implicit models as used in AVB, and deterministic encoder mappings, i.e. with .

Appendix B Proof of Theorem 1 and further details

We will consider certain sets of joint probability distributions of three random variables . The reader may wish to think of as true images, as images sampled from the model, and as latent codes. We denote by a joint distribution of a variable pair , where is first sampled from and next from . Note that defined in (3) and used throughout this work is the marginal distribution of when .

In the optimal transport problem (1), we consider joint distributions which are called couplings between values of and . Because of the marginal constraint, we can write and we can consider as a non-deterministic mapping from to . Theorem 1. shows how to factor this mapping through , i.e., decompose it into an encoding distribution and the generating distribution .

As in Section 2.2, denotes the set of all joint distributions of with marginals , and likewise for . The set of all joint distributions of such that , , and will be denoted by . Finally, we denote by and the sets of marginals on and (respectively) induced by distributions in . Note that , , and depend on the choice of conditional distributions , while does not. In fact, it is easy to check that . From the definitions it is clear that and we immediately get the following upper bound:

(9)

If  are Dirac measures (i.e., ), it turns out that :

Lemma 2.

with identity if 777We conjecture that this is also a necessary condition. The necessity is not used in the paper. are Dirac for all .

Proof.

The first assertion is obvious. To prove the identity, note that when is a deterministic function of , for any in the sigma-algebra induced by we have . This implies and concludes the proof. ∎

We are now in place to prove Theorem 1. Lemma 2 obviously leads to

The tower rule of expectation, and the conditional independence property of implies

It remains to notice that as stated earlier.

b.1 Random decoders

If the decoders are non-deterministic, Lemma 2 provides only the inclusion of sets and we get the following upper bound on the OT:

Corollary 3.

Let and assume the conditional distributions have mean values and marginal variances for all , where . Take . Then

(10)
Proof.

First inequality follows from (9). For the identity we proceed similarly to the proof of Theorem 1 and write

(11)

Note that

Together with (11) and the fact that this concludes the proof. ∎

Appendix C Further details on experiments

c.1 Mnist

We use mini-batches of size 100 and trained the models for 100 epochs. We used

and . For the encoder-decoder pair we set for Adam in the beginning and for the adversary in WAE-GAN to . After 30 epochs we decreased both by factor of 2, and after first 50 epochs further by factor of 5.

Both encoder and decoder used fully convolutional architectures with 4x4 convolutional filters.

Encoder architecture:

Decoder architecture:

Adversary architecture for WAE-GAN:

Here stands for a convolution with filters,

for the fractional strided convolution with

filters (first two of them were doubling the resolution, the third one kept it constant), for the batch normalization,

for the rectified linear units, and

for the fully connected layer mapping to . All the convolutions in the encoder used vertical and horizontal strides 2 and SAMEpadding.

Finally, we used two heuristics. First, we always pretrained separately the encoder for several mini-batch steps before the main training stage so that the sample mean and covariance of would try to match those of . Second, while training we were adding a pixel-wise Gaussian noise truncated at to all the images before feeding them to the encoder, which was meant to make the encoders random. We played with all possible ways of combining these two heuristics and noticed that together they result in slightly (almost negligibly) better results compared to using only one or none of them.

Our VAE model used cross-entropy loss (Bernoulli decoder) and otherwise same architectures and hyperparameters as listed above.

c.2 CelebA

We pre-processed CelebA images by first taking a 140x140 center crops and then resizing to the 64x64 resolution. We used mini-batches of size 100 and trained the models for various number of epochs (up to 250). All reported WAE models were trained for 55 epochs and VAE for 68 epochs. For WAE-MMD we used and for WAE-GAN . Both used .

For WAE-MMD the learning rate of Adam was initially set to . For WAE-GAN the learning rate of Adam for the encoder-decoder pair was initially set to and for the adversary to