1 Introduction
Generative Adversarial Networks (GANs) are one of the most popular generative models gan
. A neural network, the
generator, learns a map that takes random input noise to samples from a given distribution. The training involves using a second neural network, the critic, to discriminate between real samples and the generator output.In particular, wgan ; improvedWgan introduces a critic built around the Wasserstein distance between the distribution of true images and generated images. The Wasserstein distance is inherently based on a notion of distance between images which in all implementations of Wasserstein GANs (WGAN) so far has been the
distance. On the other hand, the imaging literature contains a wide range of metrics used to compare images
ImageProcessing that each emphasize different features of interest, such as edges or to more accurately approximate human observer perception of the generated image.There is hence an untapped potential in selecting a norm beyond simply the classical norm. We could for example select an appropriate Sobolev space to either emphasize edges, or large scale behavior. In this work we extend the classical WGAN theory to work on these and more general Banach spaces.
Our contributions are as follows:

We introduce Banach Wasserstein GAN (BWGAN), extending WGAN implemented via a gradient penalty (GP) term to any separable complete normed space.

We describe how BWGAN can be efficiently implemented. The only practical difference from classical WGAN with gradient penalty is that the
norm is replaced with a dual norm. We also give theoretically grounded heuristics for the choice of regularization parameters.

We compare BWGAN with different norms on the CIFAR10 and CelebA datasets. Using the Space
, which puts strong emphasize on outliers, we achieve an unsupervised inception score of
on CIFAR10, state of the art for nonprogressive growing GANs.
2 Background
2.1 Generative adversarial networks
Generative Adversarial Networks (GANs) gan perform generative modeling by learning a map from a lowdimensional latent space to image space , mapping a fixed noise distribution to a distribution of generated images .
In order to train the generative model , a second network is used to discriminate between original images drawn from a distribution of real images and images drawn from . The generator is trained to output images that are conceived to be realistic by the critic . The process is iterated, leading to the famous minimax game gan between generator and critic
(1) 
Assuming the discriminator is perfectly trained, this gives rise to the Jensen–Shannon divergence (JSD) as distance measure between the distributions and (gan, , Theorem 1).
2.2 Wasserstein metrics
To overcome undesirable behavior of the JSD in the presence of singular measures principledGANs , in wgan the Wasserstein metric is introduced to quantify the distance between the distributions and . While the JSD is a strong metric, measuring distances pointwise, the Wasserstein distance is a weak metric, measuring the cost of transporting one probability distribution to another. This allows it to stay finite and provide meaningful gradients to the generator even when the measures are mutually singular.
In a rather general form, the Wasserstein metric takes into account an underlying metric on a Polish (e.g. separable completely metrizable) space . In its primal formulation, the Wasserstein, , distance is defined as
(2) 
where denotes the set of distributions on with marginals and . The Wasserstein distance is hence highly dependent on the choice of metric .
The KantorovichRubinstein duality (villani, , 5.10) provides a way of more efficiently computing the Wasserstein1 distance (which we will henceforth simply call the Wasserstein distance, ) between measures on high dimensional spaces. The duality holds in the general setting of Polish spaces and states that
(3) 
The supremum is taken over all Lipschitz continuous functions with Lipschitz constant equal or less than one. We note that in this dual formulation, the dependence of on the choice of metric is encoded in the condition of being 1Lipschitz and recall that a function is Lipschitz if
2.3 Wasserstein GAN
Implementing GANs with the Wasserstein metric requires to approximate the supremum in (3) with a neural network. In order to do so, the Lipschitz constraint has to be enforced on the network. In the paper Wasserstein GAN wgan this was achieved by restricting all network parameters to lie within a predefined interval. This technique typically guarantees that the network is Lipschitz for some for any metric space. However, it typically reduces the set of admissible functions to a proper subset of all Lipschitz functions, hence introducing an uncontrollable additional constraint on the network. This can lead to training instabilities and artifacts in practice improvedWgan .
In improvedWgan strong evidence was presented that the condition can better be enforced by working with another characterization of Lipschitz functions. In particular, they prove that if we have the gradient characterization
They softly enforce this condition by adding a penalty term to the loss function of
that takes the form(4) 
where the distribution of
is taken to be the uniform distributions on lines connecting points drawn from
and .However, penalizing the norm of the gradient corresponds specifically to choosing the norm as underlying distance measure on image space. Some research has been done on generalizing GAN theory to other spaces FOGAN ; ApproximationsGAN , but in its current form WGAN with gradient penalty does not extend to arbitrary choices of underlying spaces . We shall give a generalization to a large class of spaces, the (separable) Banach spaces, but first we must introduce some notation.
2.4 Banach spaces of images
A vector space is a collection of objects (vectors) that can be added together and scaled, and can be seen as a generalization of the Euclidean space
. If a vector space is equipped with a notion of length, a norm , we call it a normed space. The most commonly used norm is the norm defined on , given bySuch spaces can be used to model images in a very general fashion. In a pixelized model, the image space is given by the discrete pixel values, . Continuous image models that do not rely on the concept of pixel discretization include the space of square integrable functions over the unit square. The norm gives room for a choice on how distances between images are measured. The Euclidean distance is a common choice, but many other distance notions are possible that account for more specific image features, like the position of edges in Sobolev norms.
A normed space is called a Banach space if it is complete, that is, Cauchy sequences converge. Finally, a space is separable if there exists some countable dense subset. Completeness is required in order to ensure that the space is rich enough for us to define limits whereas separability is necessary for the usual notions of probability to hold. These technical requirements formally hold in finite dimensions but are needed in the infinite dimensional setting. We note that all separable Banach spaces are Polish spaces and we can hence define Wasserstein metrics on them using the induced metric .
For any Banach space , we can consider the space of all bounded linear functionals , which we will denote and call the (topological) dual of . It can be shown RudinFA that this space is itself a Banach space with norm given by
(5) 
In what follows, we will give some examples of Banach spaces along with explicit characterizations of their duals. We will give the characterizations in continuum, but they are also Banach spaces in their discretized (finite dimensional) forms.
spaces.
Let be some domain, for example to model square images. The set of functions with norm
(6) 
is a Banach space with dual where . In particular, we note that . The parameter controls the emphasis on outliers, with higher values corresponding to a stronger focus on outliers. In the extreme case , the norm is known to induce sparsity, ensuring that all but a small amount of pixels are set to the correct values.
Sobolev spaces.
Let be some domain, then the set of functions with norm
where is the spatial gradient, is an example of a Sobolev space. In this space, more emphasis is put on the edges than in e.g. spaces, since if is small then not only are their absolute values close, but so are their edges.
Since taking the gradient is equivalent to multiplying with in the Fourier space, the concept of Sobolev spaces can be generalized to arbitrary (real) derivative orders if we use the norm
(7) 
where
is the Fourier transform. The tuning parameter
allows to control which frequencies of an image are emphasized: A negative value of corresponds to amplifying low frequencies, hence prioritizing the global structure of the image. On the other hand, high values of amplify high frequencies, thus putting emphasis on sharp local structures, like the edges or ridges of an image.The dual of the Sobolev space, , is where is as above brezis . Under weak assumptions on , all Sobolev spaces with are separable. We note that and in particular we recover as an important special case .
There is a wide range of other norms that can be defined for images, see appendix A and EncyclopediaofDistances ; brezis for a further overview of norms and their respective duals.
3 Banach Wasserstein GANs
In this section we generalize the loss (4) to separable Banach spaces, allowing us to effectively train a Wasserstein GAN using arbitrary norms.
We will show that the characterization of Lipschitz functions via the norm of the differential can be extended from the setting in (4) to arbitrary Banach spaces by considering the gradient as an element in the dual of . In particular, for any Banach space with norm , we will derive the loss function
(8) 
where are regularization parameters, and show that a minimizer of this this is an approximation to the Wasserstein distance on .
3.1 Enforcing the Lipschitz constraint in Banach spaces
Throughout this chapter, let denote a Banach space with norm and a continuous function. We require a more general notion of gradient: The function is called Fréchet differentiable at if there is a bounded linear map such that
(9) 
The differential is hence an element of the dual space . We note that the usual notion of gradient in with the standard inner product is connected to the Fréchet derivative via .
The following theorem allows us to characterize all Lipschitz continuous functions according to the dual norm of the Fréchet derivative.
Lemma 1.
Assume is Fréchet differentiable. Then is Lipschitz if and only if
(10) 
Proof.
Now let satisfy (10) and let . Define the function by
As , we see that is everywhere differentiable and
Hence
which gives
thus finishing the proof. ∎
Using lemma 1 we see that a Lipschitz requirement in Banach spaces is equivalent to the dual norm of the Fréchet derivative being less than everywhere. In order to enforce this we need to compute . As shown in section 2.4, the dual norm can be readily computed for a range of interesting Banach spaces, but we also need to compute , preferably using readily available automatic differentiation software. However, such software can typically only compute derivatives in with the standard norm.
Consider a finite dimensional Banach space equipped by any norm . By Lemma 1, gradient norm penalization requires characterizing (e.g. giving a basis for) the dual of . This can be a difficult for infinite dimensional Banach spaces. In a finite dimensional however setting, there is an linear continuous bijection given by
(11) 
This isomorphism implicitly relies on the fact that a basis of can be chosen and can be mapped to the corresponding dual basis. This does not generalize to the infinite dimensional setting, but we hope that this is not a very limiting assumption in practice.
We note that we can write where and automatic differentiation can be used to compute the derivative
efficiently. Further, note that the chain rule yields
where is the adjoint of which is readily shown to be as simple as , . This shows that computing derivatives in finite dimensional Banach spaces can be done using standard automatic differentiation libraries with only some formal mathematical corrections. In an implementation, the operators would be implicit.
In terms of computational costs, the difference between general Banach Wasserstein GANs and the ones based on the
metric lies in the computation of the gradient of the dual norm. By the chain rule, any computational step outside the calculation of this gradient is the same for any choice of underlying notion of distance. This in particular includes any forward pass or backpropagation step through the layers of the network used as discriminator. If there is an efficient framework available to compute the gradient of the dual norm, as in the case of the Fourier transform used for Sobolev spaces, the computational expenses hence stay essentially the same independent of the choice of norm.
3.2 Regularization parameter choices
The network will be trained by adding the regularization term
Here, is a regularization constant and is a scaling factor controlling which norm we compute. In particular will approximate times the Wasserstein distance. In the original WGANGP paper improvedWgan and most following work and , while was used in Progressive GAN progressiveGAN . However, it is easy to see that these values are specific to the norm and that we would need to retune them if we change the norm. In order to avoid having to handtune these for every choice of norm, we will derive some heuristic parameter choice rules that work with any norm.
For our heuristic, we will start by assuming that the generator is the zerogenerator, always returning zero. Assuming symmetry of the distribution of true images , the discriminator will then essentially be decided by a single constant , where solves the optimization problem
By solving this explicitly we find
Since we are trying to approximate times the Wasserstein distance, and since the norm has Lipschitz constant 1, we want . Hence to get a small relative error we need . With this theory to guide us, we can make the heuristic rule
In the special case of CIFAR10 with the norm this gives , which agrees with earlier practice () reasonably well.
Further, in order to keep the training stable we assume that the network should be approximately scale preserving. Since the operation is the deepest part of the network (twice the depth as the forward evaluation), we will enforce . Assuming was appropriately chosen, we find in general (by lemma 1) . Hence we want . We pick the expected value as a representative and hence we obtain the heuristic
For CIFAR10 with the norm this gives and may explain the improved performance obtained in progressiveGAN .
A nice property of the above parameter choice rules is that they can be used with any underlying norm. By using these parameter choice rules we avoid the issue of hand tuning further parameters when training using different norms.
4 Computational results
To demonstrate computational feasibility and to show how the choice of norm can impact the trained generator, we implemented Banach Wasserstein GAN with various Sobolev and norms, applied to CIFAR10 and CelebA (
pixels). The implementation was done in TensorFlow and the architecture used was a faithful reimplementation of the residual architecture used in
improvedWgan , see appendix B. For the loss function, we used the loss eq. 8 with parameters according to section 3.2 and the norm chosen according to either the Sobolev norm eq. 7 or the norm eq. 6. In the case of the Sobolev norm, we selected units such that . Following progressiveGAN , we add a small term to the discriminator loss to stop it from drifting during the training.For training we used the Adam optimizer Adam with learning rate decaying linearly from to over iterations with , . We used 5 discriminator updates per generator update. The batch size used was 64. In order to evaluate the reproducibility of the results on CIFAR10, we followed this up by training an ensemble of 5 generators using SGD with warm restarts following SGDR . Each warm restart used generator steps. Our implementation is available online^{1}^{1}1https://github.com/adlerj/bwgan.
Some representative samples from the generator on both datasets can be seen in figs. 6 and 4. See appendix C for samples from each of the and spaces investigated as well as samples from the corresponding Fréchet derivatives.
For evaluation, we report Fréchet Inception Distance (FID)FID and Inception scores, both computed from 50K samples. A high image quality corresponds to high Inception and low FID scores. On the CIFAR10 dataset, both FID and inception scores indicate that negative and large values of lead to better image quality. On CelebA, the best FID scores are obtained for values of between and and around , whereas the training become unstable for . We further compare our CIFAR10 results in terms of Inception scores to existing methods, see table 4. To the best of our knowledge, the inception score of , achieved using the space, is state of the art for nonprogressive growing methods. Our FID scores are also highly competitive, for CIFAR10 we achieve using . We also note that our result for is slightly better than the reference implementation, despite using the same network. We suspect that this is due to our improved parameter choices.
5 How about metric spaces?
Gradient norm penalization according to lemma 1 is only valid in Banach spaces, but a natural alternative to penalizing gradient norms is to enforce the Lipschitz condition directly (see regularizationWGAN ). This would potentially allow training Wasserstein GAN on general metric spaces by adding a penalty term of the form
(12) 
While theoretically equivalent to gradient norm penalization when the distributions of and
are chosen appropriately, this term is very likely to have considerably higher variance in practice.
For example, if we assume that is not bounded from below and consider two points that are sufficiently close then a penalty term of the Lipschitz quotient as in (12) imposes a condition on the differential around and in the direction only, i.e. only is ensured. In the case of two distributions that are already close, we will with high probability sample the difference quotient in a spatial direction that is parallel to the data, hence not exhausting the Lipschitz constraint, i.e. . Difference quotient penalization (12) then does not effectively enforce the Lipschitz condition. Gradient norm penalization on the other hand ensures this condition in all spatial directions simultaneously by considering the dual norm of the differential.
On the other hand, if is bounded from below the above argument fails. For example, Wasserstein GAN over a space equipped with the trivial metric
approximates the Total Variation distance villani . Using the regularizer eq. 12 we get a slight variation of Least Squares GAN lsgan . We do not further investigate this line of reasoning.
6 Conclusion
We analyzed the dependence of Wasserstein GANs (WGANs) on the notion of distance between images and showed how choosing distances other than the
metric can be used to make WGANs focus on particular image features of interest. We introduced a generalization of WGANs with gradient norm penalization to Banach spaces, allowing to easily implement WGANs for a wide range of underlying norms on images. This opens up a new degree of freedom to design the algorithm to account for the image features relevant in a specific application.
On the CIFAR10 and CelebA dataset, we demonstrated the impact a change in norm has on model performance. In particular, we computed FID scores for Banach Wasserstein GANs using different Sobolev spaces and found a correlation between the values of both and with model performance.
While this work was motivated by images, the theory is general and can be applied to data in any normed space. In the future, we hope that practitioners take a step back and ask themselves if the metric is really the best measure of fit, or if some other metric better emphasize what they want to achieve with their generative model.
Acknowledgments
The authors would like to acknowledge Peter Maass for brining us together as well as important support from Ozan Öktem, Axel Ringh, Johan Karlsson, Jens Sjölund, Sam Power and Carola Schönlieb.
The work by J.A. was supported by the Swedish Foundation of Strategic Research grants AM130049, ID140055 and Elekta. The work by S.L. was supported by the EPSRC grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, and the Cambridge Centre for Analysis. We also acknowledge the support of the Cantab Capital Institute for the Mathematics of Information.
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Appendix A Some further Banach spaces
There is some algebra for how to form new Banach spaces from known spaces. Specifically we have the following constructions that the reader might find useful.
Weighted spaces.
Let be some separable Banach space with norm , then we can construct another space with norm
where is a continuous linear bijection. It is straightforward to show that the dual space has norm
where is the adjoint of the inverse of . These weighted spaces could be used to focus on some feature of interest, e.g. focus especially on the red color channel or on some spatial region of the image, the center perhaps, that is more important.
Product spaces.
Let be Banach spaces and let be the product space with norm
then the dual space has norm
where . These spaces could be used to explicitly model the color channels or even to model multimodal data such as a generator outputting both an image and a caption.
Appendix B Network details
The implementation on CIFAR10 faithfully follows the source code from [7]
. It uses of residual blocks consisting of "nonlinearity + conv + nonlinearity + conv + residual connection" and meanpooling/nearest neighbor interpolation as building blocks. The generator starts from a latent space of 128 normally distributed random numbers and applies a dense layer to 4x4 images and applies a residual block then an interpolation repeatedly until the resolution 32x32 is reached. Then, a nonlinearity followed by a 1x1 convolution with 3 output channels is applied in order to obtain the generated color images.
The discriminator goes the other way using pooling with a final spatial meanpooling followed by a dense layer. We used ReLU nonlinearities, all convolutions uses 128 channels and we used batch normalization after the nonlinearities in the generator. Following, we used uniform He initialization for all convolutions except the residual connections which used uniform Xavier initialization.
The implementation for CelebA follows that for CIFAR10, with an additional residual block for further up/downsampling added both in the generator and discriminator.
See [7] and/or our open source implementation for further details.
Appendix C Further samples
We give samples from each of the Sobolev spaces investigated in the paper ^{2}^{2}2All images downsampled to meet size restrictions on arXiv. Full resolution version avaiable on:
https://papers.nips.cc/paper/7909banachwassersteingan. The qualitative results mirror those observed in section 4 with higher indicating higher gradients in the discriminators Fréchet derivative, thus indicating a focus on higher frequency content. We also show further examples along with the corresponding loss gradients for some spaces on both the CelebA and CIFAR10 dataset.
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