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Title: MMD GAN with Random-Forest Kernels

Paper content:
In this paper, we propose a novel kind of kernel, random forest kernel, to enhance the empirical performance of MMD GAN.
Different from common forests with deterministic routings, a probabilistic routing variant is used in our innovated random-forest kernel, which is possible to merge with the CNN frameworks.
Our proposed random-forest kernel has the following advantages: From the perspective of random forest, the output of GAN discriminator can be viewed as feature inputs to the forest, where each tree gets access to merely a fraction of the features, and thus the entire forest benefits from ensemble learning.
In the aspect of kernel method, random-forest kernel is proved to be characteristic, and therefore suitable for the MMD structure.
Besides, being an asymmetric kernel, our random-forest kernel is much more flexible, in terms of capturing the differences between distributions.
Sharing the advantages of CNN, kernel method, and ensemble learning, our random-forest kernel based MMD GAN obtains desirable empirical performances on CIFAR-10, CelebA and LSUN bedroom data sets.
Furthermore, for the sake of completeness, we also put forward comprehensive theoretical analysis to support our experimental results.
Generative adversarial nets (GANs; Goodfellow et al., 2014) are well-known generative models, which largely attribute to the sophisticated design of a generator and a discriminator which are trained jointly in an adversarial fashion.
Nowadays GANs are intensely used in a variety of practical tasks, such as image-to-image translation (Tang et al., 2019; Mo et al., 2019) ; 3D reconstruction (Gecer et al., 2019) ; video prediction (Kwon & Park, 2019) ; text-to-image generation (Zhu et al., 2019) ; just to name a few.
However, it's well-known that the training of GANs is a little tricky, see e.g. (Salimans et al., 2016) .
One reason of instability of GAN training lies in the distance used in discriminator to measure the divergence between the generated distribution and the target distribution.
For instance, concerning with the Jensen-Shannon divergence based GANs proposed in Goodfellow et al. (2014) , points out that if the generated distribution and the target distribution are supported on manifolds where the measure of intersection is zero, Jensen-Shannon divergence will be constant and the KL divergences be infinite.
Consequently, the generator fails to obtain enough useful gradient to update, which undermines GAN training.
Moreover, two non-overlapping distributions may be judged to be quite different by the Jensen-Shannon divergence, even if they are nearby with high probability.
As a result, to better measure the difference between two distributions, Integral Probability Metrics (IPM) based GANs have been proposed.
For instance, utilizes Wasserstein distance in GAN discriminator, while Li et al. (2017) adopts maximum mean discrepancy (MMD), managing to project and discriminate data in reproducing kernel Hilbert space (RKHS).
To mention, the RKHS with characteristic kernels including Gaussian RBF kernel (Li et al., 2017) and rational quadratic kernel (Bińkowski et al., 2018) has strong power in the discrimination of two distributions, see e.g. (Sriperumbudur et al., 2010) .
In this paper, inspired by non-linear discriminating power of decision forests, we propose a new type of kernel named random-forest kernel to improve the performance of MMD GAN discriminator.
In order to fit with back-propagation training procedure, we borrow the decision forest model with stochastic and differentiable decision trees from Kontschieder et al. (2015) in our random-forest kernel.
To be specific, each dimension of the GAN discriminator outputs is randomly connected to one internal node of a soft decision forest, serving as the candidate to-be-split dimension.
Then, the tree is split with a soft decision function through a probabilistic routing.
Other than the typical decision forest used in classification tasks where the value of each leaf node is a label, the leaf value of our random forest is the probability of a sample x i falling into a certain leaf node of the forest.
If the output of the discriminator is denoted as h θ N (x i ) and the probability output of the t-th tree is denoted as µ t (h θ N (x i ); θ F ), the random forest kernel k RF can be formulated as
where T is the total number of trees in the forest, θ N and θ F denote the parameters of the GAN discriminator and the random forest respectively.
Recall that random forest and deep neural networks are first combined in Kontschieder et al. (2015) , where differentiable decision tree model and deep convolutional networks are trained together in an end-to-end manner to solve classification tasks.
Then, Shen et al. (2017) extends the idea to label distribution learning, and Shen et al. (2018) makes further extensions in regression regime.
Moreover, Zuo & Drummond (2017) , Zuo et al. (2018) and Avraham et al. (2019) also introduce deep decision forests.
Apart from the typical ensemble method that averages the results across trees, they aggregate the results by multiplication.
As for the combination of random forest and GAN, Zuo et al. (2018) introduce forests structure in GAN discriminator, combining CNN network and forest as a composited classifier, while Avraham et al. (2019) uses forest structure as one of non-linear mapping functions in regularization part.
On the other hand, in the aspect of relationship between random forest and kernel method, Breiman (2000) initiates the literature concerning the link.
He shows the fact that a purely random tree partition is equivalent to a kernel acting on the true margin, of which form can be viewed as the probability of two samples falling into the same terminal node.
Shen & Vogelstein (2018) proves that random forest kernel is characteristic.
Some more theoretical analysis can be found in Davies & Ghahramani (2014) , Arlot & Genuer (2014) , Scornet (2016) .
However, despite their theoretical breakthroughs, forest decision functions used in these forest kernels are non-differentiable hard margins rather than differentiable soft ones, and thus cannot be directly used in back propagation regime.
To the best of our knowledge, MMD GAN with our proposed random-forest kernel is the first to combine random forest with deep neural network in the form of kernel MMD GAN.
Through theoretical analysis and numerical experiments, we evaluate the effectiveness of MMD GAN with our random-forest kernel.
From the theoretical point of view, our random-forest kernel enjoys the property of being characteristic, and the gradient estimators used in the training process of random-forest kernel GAN are unbiased.
In numerical experiments, we evaluate our random-forest kernel under the setting of both the original MMD GAN (Li et al., 2017) and the one with repulsive loss (Wang et al., 2019) .
Besides, we also compare our random-forest kernel with Gaussian RBF kernel (Li et al., 2017) , rational quadratic kernel (Bińkowski et al., 2018) , and bounded RBF kernel (Wang et al., 2019) .
As a result, MMD GAN with our random-forest kernel outperforms its counterparts with respect to both accuracy and training stability.
This paper is organized as follows.
First of all, we introduce some preliminaries of MMD GAN in Section 2.
Then we review the concept of deep random forest and show how it is embedded within a CNN in 3.1.
After that, random-forest kernels and MMD GAN with random-forest kernels are proposed in 3.2 and 3.3 respectively.
Besides, the training techniques of MMD GAN with random-forest kernel are demonstrated in Section 3.4 and the theoretical results are shown in Section 3.5.
Eventually, Section 4 presents the experimental setups and results, including the comparison between our proposed random-forest kernel and other kernels.
In addition, all detailed theoretical proofs are included in the Appendices.
The generative model captures the data distribution P X , by building a mapping function G : Z → X from a prior noise distribution P Z to data space.
While the discriminative model D : X → R is used to distinguish generated distribution P Y from real data distribution P X .
Taking X, X ∼ P X and Y, Y ∼ P Y := P G (Z) where Y := G(Z) and Y := G(Z ), the squared MMD is expressed as
The loss of generator and discriminator in MMD GAN proposed in Li et al. (2017) is:
Wang et al. (2019) proposed MMD GAN with repulsive loss, where the objective functions for G and D are:
we can write an unbiased estimator of the squared MMD in terms of k as
When k is a characteristic kernel, we have MMD 2 [P X , P Y ] ≥ 0 with equality applies if and only if P X = P Y .
The best-known characteristic kernels are gaussian RBF kernel and rational quadratic kernel (Bińkowski et al., 2018) .

assistant: text: Equip MMD GANs with a new random-forest kernel.