Original:  L EARNING D IVERSE G ENERATIONS USING D ETERMINANTAL P OINT P ROCESSES
Revised: GDPP: L EARNING D IVERSE G ENERATIONS USING D ETERMINANTAL P OINT P ROCESSES

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Generative models have proven to be an outstanding tool for representing highdimensional probability distributions and generating realistic looking images.
Revised: Generative models have proven to be an outstanding tool for representing highdimensional probability distributions and generating realistic looking images.

Original: A fundamental characteristic of generative models is their ability to produce multimodal outputs.
Revised: A fundamental characteristic of generative models is their ability to produce multimodal outputs.

Original: However while training, they are often susceptible to mode collapse, which means that the model is limited in mapping the input noise to only a few modes of the true data distribution.
Revised: However, while training, they are often susceptible to mode collapse, which means that the model is limited in mapping the input noise to only a few modes of the true data distribution.

Original: In this paper, we draw inspiration from
Revised: In this paper, we draw inspiration from

Original: Determinantal Point Process (DPP) to devise a generative model that alleviates mode collapse problem while producing higher quality samples.
Revised: Determinantal Point Process (DPP) to devise a generative model that alleviates mode collapse problem while producing higher quality samples.

Original: DPP is an elegant probabilistic measure used to model negative correlations within a subset, and hence quantify its diversity.
Revised: DPP is an elegant probabilistic measure used to model negative correlations within a subset and hence quantify its diversity.

Original: We propose a generation penalty term that encouragesthe generator to behave as a Determinantal Point Process sampler and hence learns to generates diverse data.
Revised: We propose a generation penalty term that encourages generator to behave as a Determinantal Point Process sampler and hence learns generating diverse data.

Original: In contrast to previous state-of-the-art generative models that tend to use additional trainable parameters or complex training paradigms, our method does not change the original training scheme.
Revised: In contrast to previous state-of-the-art generative models that tend to use additional trainable parameters or complex training paradigms, our method does not change the original training scheme.

Original: Embedded in an adversarial strategy, our Generative DPP approach shows a consistent resistance to mode-collapse on a wide-variety of synthetic data and natural image datasets including MNIST and CIFAR10, while outperforming state-of-the-art methods for data-efﬁciency, convergence-time, and generation quality.
Revised: Embedded in an adversarial training and variational autoencoder, our Generative DPP approach shows a consistent resistance to mode-collapse on a wide-variety of synthetic data and natural image datasets including MNIST and CIFAR10, while outperforming stateof-the-art methods for data-efﬁciency, convergence-time, and generation quality.

Original: Our code will be made publicly available.
Revised: Our code will be made publicly available.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Deep generative models have gained enormous research interest inthe recent years as a powerful framework to learn high dimensional data in  unsupervised fashion.
Revised: Deep generative models have gained enormous research interest in recent years as a powerful framework to learn high dimensional data in an unsupervised fashion.

Original: Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) are among the most dominant generative approaches, and consist of training two networks: a generator and a discriminator.
Revised: Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) are among the most dominant generative approaches, and consist of training two networks: a generator and a discriminator.

Original: The generator attempts to map random noise to fake data points that simulate the probability distribution of real data.
Revised: The generator attempts to map random noise to fake data points that simulate the probability distribution of real data.

Original: In adversary, the discriminator aims to distinguish real data from fake data.
Revised: In adversary, the discriminator aims to distinguish real data from fake data.

Original: The constant competition between the networks results in improved performance of both of them.
Revised: The constant competition between the networks results in improved performance of both of them.

Original: GANs are typically attributed with high quality and sharp images.
Revised: GANs are typically attributed with high quality and sharp images.

Original: Nevertheless, in the process of learning such multi-modal complex distribution, the system may converge to a trivial solution where the generator only learns to produce few modes exclusively, as referred as the mode collapse problem.
Revised: Nevertheless, in the process of learning such multi-modal complex distribution, the system may converge to a trivial solution where the generator only learns to produce few modes exclusively, as referred as the mode collapse problem.

Original: To address this, we propose to model the generator as a Determinantal Point Process (DPP) sampler hence it learns to generates diverse examples.
Revised: To address this, we propose to model the generator as a Determinantal Point Process (DPP) sampler hence it learns to generates diverse examples.

Original: DPP is a probabilistic model that has been mainly adopted to solve subset selection problems with diversity constraints (Kulesza & Taskar, 2011), such as video and document summarization.
Revised: DPP is a probabilistic model that has been mainly adopted to solve subset selection problems with diversity constraints (Kulesza & Taskar, 2011), such as video and document summarization.

Original: In these application, DPPs maximizes the diversity selected subset to be diverse.

Original: From generation perspective, we instead start with generator G suffers from mode collapse and we aim at teaching it to generate a subset or all possible generations with diversity constraints.
Revised: From generation perspective, we instead start with generator G suffers from mode collapse and we aim at teaching it to generate a subset or all possible generations with diversity constraints.

Original: The key idea is to design a diversity loss function that encourages the generator to behave as a DPP sampler with a simple and tractable approach.
Revised: The key idea is to design a diversity loss function that encourages the generator to behave as a DPP sampler with a simple and tractable approach.

Original: By attempting to force the generator on producing samples of similar diversity distribution to the true-data distribution, the generator learns to explicitly cover signiﬁcantly more modes of the real distributions alleviating the mode collapse problem, see Fig. 1.
Revised: By attempting to force the generator on producing samples of similar diversity distribution to the true-data distribution, the generator learns to explicitly cover signiﬁcantly more modes of the real distributions alleviating the mode collapse problem, see Fig. 1.

Original: Recent approaches tackled the mode-collapse problem in one of two different ways: (1) improving the learning of the system to reach a better convergence point(e.g.
Revised: Recent approaches tackled the mode-collapse problem in one of two different ways: (1) improving the learning of the system to reach a better convergence point(e.g.

Original: Metz et al.
Revised: Metz et al.

Original: ; Arjovsky &  p 1 ppp d a t a ( x ) p 1 ppp 0 ( z ) p 1 pp 3 p ( z ) w it h D
Revised: ; Arjovsky & Bottou (2017)); or (2) explicitly enforcing the models to capture diverse modes or map back to the p 1 ppp d a t a ( x ) p 1 ppp 0 ( z ) p 1 pp 3 p ( z ) w it h D

Original: PP
Revised: PP

Original: [Figure]
Revised: [Figure]

Revised: ( Y = { p

Revised: [Figure]

Original: Bottou (2017)); or (2) explicitly enforcing the models to capture diverse modes or map back to the true-data distribution (e.g. Srivastava et al.(2017);
Revised: p 3 } ) > P ( Y = { p 1 , p 2 } ) [Figure] true-data distribution (e.g. Srivastava et al.

Original:  Che et al.
Revised: ; Che et al.

Original: Here we focus on a relaxed version of the former, where we use the same learning paradigm of the standard GANs and only change the objective function.
Revised: Here we focus on a relaxed version of the former, where we use the same learning paradigm of the standard GANs and only change the objective function.

Original: The advantage of such an approach is to avoid adding any extra trainable parameters to the trained system while maintaining the same back-propagation steps as the standard GANs.
Revised: The advantage of such an approach is to avoid adding any extra trainable parameters to the trained system while maintaining the same back-propagation steps as the standard GANs.

Original: Thus, our model converges faster to a fair equilibrium point where the generator captures the diversity of the true-data distribution while preserving the quality of the generations.
Revised: Thus, our model converges faster to a fair equilibrium point where the generator captures the diversity of the true-data distribution while preserving the quality of the generations.

Original: Contribution .
Revised: Contribution .

Original: We introduce a new loss function, we denote as Generative Determinantal Point
Revised: We introduce a new loss function, we denote as Generative Determinantal Point

Original: Processes ( GDPP ) loss.
Revised: Processes ( GDPP ) loss.

Original: Our loss only assumes an access to a Generator G , a feature extraction function φ ( · ) , and sampler from true data distribution p d .
Revised: Our loss only assumes an access to a Generator G , a feature extraction function φ ( · ) , and sampler from true data distribution p d .

Original: The loss encourages the generator G to behave as a DPP sampler producing diverse examples.
Revised: The loss encourages the generator G to behave as a DPP sampler producing diverse examples.

Original: This criterion can be considered as a complement to the original adversarial loss which attempts to learn an indistinguishable distribution from the true-data distribution without being speciﬁc to diverse modes.
Revised: This criterion can be considered as a complement to the original adversarial loss which attempts to learn an indistinguishable distribution from the true-data distribution without being speciﬁc to diverse modes.

Original: We assess the performance of GDPP on three different synthetic data environments, while also verifying the superiority on two real-world images datasets.
Revised: We assess the performance of GDPP on three different synthetic data environments, while also verifying the superiority on two real-world images datasets.

Original: We compared our approach with state-of-the-art approaches of more complex architectures and learning paradigms and our experiments show that outperforms all the competing methods in terms of alleviating mode-collapse, and the generator quality.
Revised: We compared our approach with state-of-the-art approaches of more complex architectures and learning paradigms and our experiments show that outperforms all the competing methods in terms of alleviating mode-collapse, and the generator quality.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Among the tremendous amount of work that tackle the training challenges of Generative Adversarial Networks (GANs), a few methods stood out as signiﬁcant contributions towards addressing the problem of mode collapse.
Revised: Among the tremendous amount of work that tackles the training challenges of Generative Adversarial Networks (GANs), a few methods stood out as signiﬁcant contributions towards addressing the problem of mode collapse.

Original: Methods that map the data back to noise.
Revised: Methods that map the data back to noise.

Original: (Donahue et al., 2017; Dumoulin et al., 2017) are one of the earliest methods that proposed learning a reconstruction network besides learning the deep generative network.
Revised: (Donahue et al., 2017; Dumoulin et al., 2017) are one of the earliest methods that proposed learning a reconstruction network besides learning the deep generative network.

Original: Adding this extra network to the system aims at reversing the action of the generator by mapping from data to noise.
Revised: Adding this extra network to the system aims at reversing the action of the generator by mapping from data to noise.

Original: Likelihood-free variational inference (LFVI)
Revised: Likelihood-free variational inference (LFVI)

Original: (Tran et al., 2017), merge this concept with learning implicit densities using hierarchical Bayesian modeling.
Revised: (Tran et al., 2017), merge this concept with learning implicit densities using hierarchical Bayesian modeling.

Original: Ultimately, VEEGAN (Srivastava et al., 2017) used the same concept, but the authors did not base their reconstruction loss on the discriminator.
Revised: Ultimately, VEEGAN (Srivastava et al., 2017) used the same concept, but the authors did not base their reconstruction loss on the discriminator.

Original: This has the advantage of isolating the generation process from the discriminator’s sensitivity to any of the modes.
Revised: This has the advantage of isolating the generation process from the discriminator’s sensitivity to any of the modes.

Original: Che et al.
Revised: Che et al.

Original: (2017) proposed several ways of regularizing the objective of adversarial learning including geometric metric regularizer, mode regularizer and manifold-diffusion training.
Revised: (2017) proposed several ways of regularizing the objective of adversarial learning including geometric metric regularizer, mode regularizer, and manifold-diffusion training.

Original: Mode regularization, speciﬁcally has shown a potential into addressing the mode collapse problem and stabilizing the GANs training in general.
Revised: Mode regularization speciﬁcally has shown a potential into addressing the mode collapse problem and stabilizing the GANs training in general.

Original: Methods that provide a surrogate objective function.
Revised: Methods that provide a surrogate objective function.

Original: Chen et al.
Revised: Chen et al.

Original: (2016) on the other hand propose with InfoGAN an information-theoretic extension of GANs that obtains disentangled representation of data by latent-code reconstitution through a penalty term in its objective function.
Revised: (2016) on the other hand propose with InfoGAN an information-theoretic extension of GANs that obtains disentangled representation of data by latent-code reconstitution through a penalty term in its objective function.

Original: InfoGAN includes an autoencoder over the latent codes; however it was shown to have stability problems similar to the standard GAN and requires stabilization tricks.
Revised: InfoGAN includes an autoencoder over the latent codes; however, it was shown to have stability problems similar to the standard GAN and requires stabilization tricks.

Original: Ghosh et al.
Revised: Ghosh et al.

Original: (2018) base the ModeGAN method on the assumption of the availability of sufﬁcient samples of every mode on the training data.
Revised: (2018) base the ModeGAN method on the assumption of the availability of sufﬁcient samples of every mode on the training data.

Original: In particular, if a sample from the true data distribution belongs to a particular mode, then the generated fake sample is likely to belong to the same mode.
Revised: In particular, if a sample from the true data distribution belongs to a particular mode, then the generated fake sample is likely to belong to the same mode.

Original: The Unrolled-GAN of Metz et al.
Revised: The Unrolled-GAN of Metz et al.

Original: (2017) propose a novel objective to update the generator with respect to the unrolled optimization of the discriminator.
Revised: (2017) propose a novel objective to update the generator with respect to the unrolled optimization of the discriminator.

Original: This allows training to be adjusted between using the optimal discriminator in the generator’s objective.
Revised: This allows training to be adjusted between using the optimal discriminator in the generator’s objective.

Original: It has been shown to improve the generator training process which in turn helps to reduce the mode collapse problem.
Revised: It has been shown to improve the generator training process which in turn helps to reduce the mode collapse problem.

Original: Generalized LS-GAN of Edraki & Qi (2018) deﬁne a pullback operator to map generated samples to the data manifold.
Revised: Generalized LS-GAN of Edraki & Qi (2018) deﬁne a pullback operator to map generated samples to the data manifold.

Original: There is however no speciﬁc enforcement of diversity of samples.
Revised: There is, however, no speciﬁc enforcement of diversity of samples.

Original: Spectral normalization strategies have been recently proposed in the works of Miyato et al.
Revised: Spectral normalization strategies have been recently proposed in the works of Miyato et al.

Original: (2018) and SAGAN (Zhang et al., 2018) to further stabilize the training.
Revised: (2018) and SAGAN (Zhang et al., 2018) to further stabilize the training.

Original: We note that these strategies are orthogonal to our contribution and could be implemented in conjunction with ours to further improve the training stability of generator models.
Revised: We note that these strategies are orthogonal to our contribution and could be implemented in conjunction with ours to further improve the training stability of generator models.

Original: Finally, improving the Wasserstein GANs of Arjovsky et al.
Revised: Finally, improving the Wasserstein GANs of Arjovsky et al.

Original: WGAN-GP (Gulrajani et al., 2017) introduce a gradient penalization employed in state-of-the-art systems (Karras et al., 2018).
Revised: WGAN-GP (Gulrajani et al., 2017) introduce a gradient penalization employed in state-of-the-art systems (Karras et al., 2018).

Original: Methods that use multiple generators.
Revised: Methods that use multiple generators.

Original: One of the popular methods to reduce mode collapse is using multiple generator networks to provide a better coverage of the true data distribution.
Revised: One of the popular methods to reduce mode collapse is using multiple generator networks to provide a better coverage of the true data distribution.

Original: Liu &
Revised: Liu &

Original: Tuzel (2016) propose using two generators with shared parameters to learn the joint distribution of the data.
Revised: Tuzel (2016) propose using two generators with shared parameters to learn the joint distribution of the data.

Original: The two generators are trained independently on two domains to ensure  diverse generation.
Revised: The two generators are trained independently on two domains to ensure a diverse generation.

Original: However, sharing the parameters guide both the generators to a similar subspace.
Revised: However, sharing the parameters guide both the generators to a similar subspace.

Original: Also, Durugkar et al.
Revised: Also, Durugkar et al.

Original: (2017) propose a similar idea of multiple discriminators that are being  ensemble, which was shown to produce better quality samples.
Revised: (2017) propose a similar idea of multiple discriminators that are being an ensemble, which was shown to produce better quality samples.

Original: Recently, Ghosh et al.
Revised: Recently, Ghosh et al.

Original: (2018) proposed MAD-GAN which is a multi-agent GAN architecture incorporating multiple generators and one discriminator.
Revised: (2018) proposed MAD-GAN which is a multi-agent GAN architecture incorporating multiple generators and one discriminator.

Original: Along with distinguishing real from fake samples, the discriminator also learns to identify the generator that generated the fake sample.
Revised: Along with distinguishing real from fake samples, the discriminator also learns to identify the generator that generated the fake sample.

Original: The learning of such  system implies forcing different generators to learn unique modes, which helps in a better coverage of data modes.
Revised: The learning of such a system implies forcing different generators to learn unique modes, which helps in a better coverage of data modes.

Original: In contrast to these approaches, our DPP-GAN does not require the training of extra networks.
Revised: In contrast to these approaches, our DPP-GAN does not require the training of extra networks.

Original: B ACKGROUND 3.  D ETERMINANTAL P OINT P ROCESS (DPP)
Revised:  D ETERMINANTAL P OINT P ROCESS (DPP)

Original: DPP is a probabilistic measure that was introduced in quantum physics( Macchi (1975)) and have been studied extensively in random matrix theory( Hough et al.
Revised: DPP is a probabilistic measure that was introduced in quantum physics( Macchi (1975)) and has been studied extensively in random matrix theory( Hough et al.

Original: It provides a tractable and efﬁcient means to capture negative correlation with respect to a similarity measure, that in turn can be used to quantify the diversity within a subset.
Revised: It provides a tractable and efﬁcient means to capture negative correlation with respect to a similarity measure, that in turn can be used to quantify the diversity within a subset.

Original: A key characteristic in DPP that the model is agnostic about the order of items as pointed out by Gong et al.
Revised: A key characteristic in DPP that the model is agnostic about the order of items as pointed out by Gong et al.

Original: (2014), and therefore can be used to model data that is randomly sampled from a certain distribution.
Revised: (2014), and therefore can be used to model data that is randomly sampled from a certain distribution.

Original: A point process P on a ground set V is a probability measure on the power set of V (i.e., 2 N ), where
Revised: A point process P on a ground set V is a probability measure on the power set of V (i.e., 2 N ), where

Original: N = |V| is the size of the ground set.
Revised: N = |V| is the size of the ground set.

Original: A point process P is called determinantal if, given a random subset Y drawn according to P , we have for every S ⊆ Y , 
Revised: A point process P is called determinantal if, given a random subset Y drawn according to P , we have for every S ⊆ Y , [Equation]

Original: [Equation]
Revised: [Equation]

Original: (1) for some symmetric similarity kernel L ∈ R N × N  .
Revised: for some symmetric similarity kernel L ∈ R N × N , where L S is the similarity kernel of subset S .

Original: L must be real, positive semideﬁnite matrix 
Revised: L must be real, positive semideﬁnite matrix L (cid:22)

Original: L (cid:22) I (all the eigenvalues of L is between 0 and 1); since it represents a probabilistic measure and all of its principal minors must be non-negative.
Revised: I (all the eigenvalues of L is between 0 and 1); since it represents a probabilistic measure and all of its principal minors must be non-negative.

Original: L is often referred to as the marginal kernel because it contains all the information needed to compute the probability of any subset S being selected in V .
Revised: L is often referred to as the marginal kernel because it contains all the information needed to compute the probability of any subset S being selected in V .

Original: L S denotes the sub-matrix of L indexed by S .
Revised: L S denotes the sub-matrix of L indexed by S .

Original: Hence, the marginal probability of including one element e i is p ( e i ∈ Y ) =
Revised: Hence, the marginal probability of including one element e i is p ( e i ∈ Y ) =

Original: L ii , and two elements e i and e j is L ii L jj − 2 L 2 ij = p
Revised: L ii , and two elements e i and e j is L ii L jj − 2 L 2 ij = p

Original: ( e i ∈ Y ) p ( e j ∈ Y ) − L 2 ij .
Revised: ( e i ∈ Y ) p ( e j ∈ Y ) − L 2 ij .

Original: A large value of L ij reduces the likelihood of both elements to appear together.
Revised: A large value of L ij reduces the likelihood of both elements to appear together.

Original: The kernel L is indexed by the elements of S , where
Revised: The kernel L is indexed by the elements of S , where

Original: L S ≡ [ L ij ]; i, j ∈ S .
Revised: L S ≡ [ L ij ]; i, j ∈ S .

Original: Kulesza & Taskar (2010) proposed decomposing the kernel L as a Gram matrix:
Revised: Kulesza & Taskar (2010) proposed decomposing the kernel L as a Gram matrix:

Original: [Equation]
Revised: [Equation]

Original: Here q ( e i ) ≥ 0 may be seen as a quality score of an item e i in the ground set V , while φ i ∈ R D ; D ≤ N and || φ i || 2 = 1 is used as an (cid:96) 2 normalized feature vector of an item.
Revised: Here q ( e i ) ≥ 0 may be seen as a quality score of an item e i in the ground set V , while φ i ∈ R D ; D ≤ N and || φ i || 2 = 1 is used as an (cid:96) 2 normalized feature vector of an item.

Original: In this manner, φ (cid:62) i φ j ∈ [ − 1 , 1] is evaluated as a ”normalized similarity” between items e i and e j of V , and the kernel L  is guaranteed to be real positive semideﬁnite matrix.Geometric interpretation. det( φ ( S ) (cid:62) φ ( S )) = (cid:81)
Revised: In this manner, φ (cid:62) i φ j ∈ [ − 1 , 1] is evaluated as a ”normalized similarity” between items e i and e j of V , and the kernel L S is guaranteed to be real positive semideﬁnite matrix.

Original: i λ i , where λ i is the i th eigen value of the matrix φ ( S ) (cid:62) φ ( S ) .

Original: Hence, we may visualize that DPP models diverse representations of data because the determinant of φ ( S ) (cid:62) φ ( S ) corresponds to the volume in n-D represented by the multiplication of the variances of the data (i.e., the eigen values).

Original: DPP in literature: DPP has proven to be a tremendously valuable tool when addressing the problem of diversity enforcing such as document summarization (e.g., Kulesza & Taskar (2011); Hong & Nenkova (2014)), pose estimation (e.g., Gupta (2015)) and video summarization (e.g., Gong et al.
Revised: DPP in literature: DPP has proven to be a tremendously valuable tool when addressing the problem of diversity enforcing such as document summarization (e.g., Kulesza & Taskar (2011); Hong & Nenkova (2014)), pose estimation (e.g., Gupta (2015)) and video summarization (e.g., Gong et al.

Original: (2014); Mahasseni et al.
Revised: (2014); Mahasseni et al.

Original: For instance, Zhang et al.
Revised: For instance, Zhang et al.

Original: (2016) proposed to learn the two parameters q, φ in Eq.
Revised: (2016) proposed to learn the two parameters q, φ in eq.

Original: 2 to quantify the diversity of the kernel L S Recently.
Revised: 2 to quantify the diversity of the kernel L S using MLPs based on spatio-temporal features of the video to perform summarization.

Original:  Hsiao & Grauman (2018) proposed to use DPP to automatically create capsule wardrobes, i.e. assemble a minimal set of items that provides maximal mix-and-match outﬁts given an inventory of candidate garments.
Revised: Recently, Hsiao & Grauman (2018) proposed to use DPP to automatically create capsule wardrobes, i.e. assemble a minimal set of items that provide maximal mix-and-match outﬁts given an inventory of candidate garments.

Original: 3.  G ENERATIVE A DVERSARIAL N ETWORKS (GAN S )
Revised:  G ENERATIVE D ETERMINANTAL P OINT P ROCESSES

Original: GANs (Goodfellow et al., 2014) learn to sample from a distribution by training two networks, a generator G and a discriminator D to compete with each other. [Equation] where x is a real image from the data distribution p data , z is a noise vector sampled from distribution p z (e.g., uniform or Gaussian distribution).
Revised: In this section, we introduce our Generative Determinantal Point Process (GDPP) loss.

Original: In practice, the generator G and thediscriminator D , are trained by sampling two batches from real images and  fake images. By optimizing D and
Revised: Our loss encourages the generator to sample fake data of diversity similar to the real data diversity. The key challenge is to model the diversity within the real data and the fake data.

Original: G alternatively, GAN is trained to generate images that emulate the training distribution p data . However, as described in the literature, a generator with a good capability to sample a few images of the real data would minimize Eq. 3,making standard GANs suffer from mode-collapse problem.  G ENERATIVE D ETERMINANTAL P OINT P ROCESSES
Revised: As discussed in Sec. 3,

Original: We now introduce our Generative Determinantal Point Processes (GDPP) loss, which encourages learning to sample data that match the diversity of the true distribution. The key idea is  to quantify the negative correlation within a discrete data distribution.
Revised: DPP is used to model the negative correlation within a discrete data distribution, which is commonly employed as a measure of diversity. Therefore, we construct a DPP kernel for both the training data and the generated samples at every iteration of the training process.

Original: For that, we utilize DPP to construct a  diversity kernel for the true data distribution as well as for the fake data distribution. Then we encourage the generator to synthesize samples with a diversity similar to  the diversity of  real data.
Revised: Then, we encourage the network to generate samples that have a similar diversity kernel to that of the training data. To simplify the learning process, we choose to match the eigenvalues and eigenvectors of the fake data DPP kernel with their corresponding of the real data DPP kernel as discussed in Fig.

Original:  During training,we have access to a generator Gthat produces a batch of samples S B = { e 1 , e 2 , · · · e B } , where B is the batch size.
Revised: 1b. During training, a generative model G produces a batch of samples S B = { e 1 , e 2 , · · · e B } , where B is the batch size.

Original: In order to have a set of diverse generations, we aim at producing S B that is probabilistically sampled with a DPP 
Revised: Our aim is to produce S B that is probabilistically sampled following the DPP which satisﬁes:

Original: [Equation]
Revised: [Equation]

Original: such that Y is a random variable that represents a subset drawn with a generative point process P , and L S B is the kernel matrix of the subset indices by S , as detailed in Sec. 3.1.
Revised: such that Y is a random variable that represents a subset drawn with a generative point process P , and L S B is the kernel matrix of the subset indices by S , as detailed in Sec. 3.

Original: Connecting DPP to the data generation, we assume that G is the point process sampler that generates the subset S B according to P .
Revised: Connecting DPP to the data generation, we assume that G is the point process sampler that generates the subset S B according to P .

Original: Let φ ( S B ) ∈
Revised: Let φ ( S B ) ∈

Original: R
Revised: R

Original: d × B be a feature representation of the generated subset S B , where φ ( · ) is some feature extraction function.
Revised: d × B be a feature representation of the generated subset S B , where φ ( · ) is some feature extraction function.

Revised: Therefore, the DPP kernel is constructed as follows:

Original: [Equation]
Revised: [Equation]

Original: where z B ∈ R d z × B is noise vector that corresponds to the  generated subset S B .
Revised: where z B ∈ R d z × B is noise vector inputted to the generator G resulting in the generated subset S B .

Revised: Let us denote φ ( S B ) a feature representation of a generated batch and φ ( D B ) a feature representation of a true batch.

Original: Formally speaking, we aim at matching P ( S B ⊆ Y )
Revised: Our aim is to match P ( S B ⊆ Y )

Original: ∝ [det( L S B ) =
Revised: ∝ [det( L S B ) =

Original: (cid:81) i λ ifake ] to P ( D B ∼ p d ) ∝ [det( L D B ) =

Original: (cid:81) 
Revised: (cid:81) i λ ifake ] to

Original: i λ i real ] , where D B is a batch sampled from the real data distribution p d  ,  λ i real and λ ifake are the i th eigenvalues of L D B and L S B respectively.
Revised: P ( D B ∼ p d ) ∝ [det( L D B ) = (cid:81) i λ ireal ] , where λ ireal and λ ifake are the i th eigenvalues of L D B and L S B respectively.

Original: Hence, the problem is reduced to learn a fake diversity kernel L S B close to the real diversity kernel L D B .
Revised: Thus, our problem is reduced to learn a fake diversity kernel L S B close to the real diversity kernel L D B .

Original: As demonstrated in Figure 1b, we choose to solve the learning problem by optimizing the eigenvalues and eigenvectors of the fake kernel to match their corresponding ones of the real kernel.
Revised: We choose to represent those kernels using their eigenvalues and eigenvectors.

Original: Our GDPP loss is composed of two components: diversity magnitude loss L m , and diversity structure loss L s as follows:
Revised: Our GDPP loss is composed of two components: diversity magnitude loss L m , and diversity structure loss L s as follows:

Original: [Equation]
Revised: [Equation]

Original: where v ifake and v ireal are the i th eigenvectors of L D B and L S B respectively.
Revised: where v ifake and v ireal are the i th eigenvectors of L D B and L S B respectively.

Original: ˆ λ ireal are the min-max normalized version of its corresponding eigenvalues λ ireal .
Revised: ˆ λ ireal are the min-max normalized version of its corresponding eigenvalues λ ireal .

Original: We note that λ ≥ 0 for both fake and real data, since both L S B and L D B are guranteed to be positive semideﬁnite matrices.
Revised: We note that λ ≥ 0 for both fake and real data, since both L S B and L D B are guaranteed to be positive semideﬁnite matrices.

Original: Scaling the structure loss aims to induce noise invariance within the eigenvectors similarity learning.
Revised: Scaling the structure loss aims to induce noise invariance within the eigenvectors similarity learning.

Original: This can be seen as alleviating the effect of outlier structures that intrinsically exist within the real data on learning the diversity kernel of fake data.
Revised: This can be seen as alleviating the effect of outlier structures that intrinsically exist within the real data on learning the diversity kernel of fake data.

Original: We note that all the eigenvalues of L S B and L S D will be real non-negatives since both of the kernels are symmetric semi-positive deﬁnite.
Revised: We note that all the eigenvalues of L S B and L S D will be real non-negatives since both of the kernels are symmetric semi-positive deﬁnite.

Original: Therefore, the kernels represent a probability measure since none of the principal minors will be negative.
Revised: Therefore, the kernels represent a probability measure since none of the principal minors will be negative.

Original: Integrating GDPP loss with GANs.
Revised: Integrating GDPP loss with GANs.

Original: We choose to integrate our GDPP loss with GANs.
Revised: For a primary benchmark, we integrate our GDPP loss with GANs.

Original: Since we aim  to avoid adding any extra trainable parameters, we utilize featuresthat are extracted by the discriminator.
Revised: Since our aim is to avoid adding any extra trainable parameters, we utilize features extracted by the discriminator.

Original: We choose to use the hidden activations before the last layer as our feature extraction function φ ( . ) .
Revised: We choose to use the hidden activations before the last layer as our feature extraction function φ ( . ) .

Original: We apply (cid:96) 2 normalization on the obtained features that guarantees constructing a positive semi-deﬁnite matrix according to eq.
Revised: We apply (cid:96) 2 normalization on the obtained features that guarantees constructing a positive semi-deﬁnite matrix according to eq.

Original: We ﬁnally integrate L DPPg into the GAN objective by only modifying the generator loss  as follows:
Revised: We ﬁnally integrate L DPPg into the GAN objective by only modifying the generator loss of the standard adversarial loss (Goodfellow et al., 2014) as follows:

Original: [Equation]
Revised: [Equation]

Revised: Integrating GDPP loss with VAEs.

Revised: A key property of our loss that it is generalizable to any generative model.

Revised: We show that by also incorporating it with Variational Auto-Encoders (VAEs) proposed in Kingma & Welling (2013).

Revised: We use the decoder network as our generator G , and the ﬁnal hidden activations within the encoder network as our feature extraction function φ ( . ) .

Revised: To compute L DPP at the training time, we feed an input training batch S D to the encoder constructing

Revised: L S D .

Revised: We also feed a random Gaussian noise to the decoder that generates a fake batch S B , which we then feed it to the encoder to construct L S B .

Revised: We ﬁnally compute L DPP as stated in eq.

Revised: and we add it to the original VAE loss at the training time as follows:

Revised: [Equation]

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In our experiments, we target evaluating the generation based on two criteria: mode collapse and generated samples quality.
Revised: In our experiments, we target evaluating the generation based on two criteria: mode collapse and generated samples quality.

Original: Due to the intractability of log-likelihood estimation, this problem tends to be non-trivial in real data.
Revised: Due to the intractability of log-likelihood estimation, this problem tends to be non-trivial in real data.

Original: Therefore, we start by analyzing our method on synthetic data where we can accurately evaluate the performance.
Revised: Therefore, we start by analyzing our method on synthetic data where we can accurately evaluate the performance.

Original: Then, we demonstrate the effectiveness of our method on real data using standard evaluation metrics.
Revised: Then, we demonstrate the effectiveness of our method on real data using standard evaluation metrics.

Original: We note that the same architecture is used for all the methods on the same data; one for synthetic data and one for the real datasets (See appendix A for details).
Revised: We note that the same architecture is used for all the methods on the same data; one for synthetic data and one for the real datasets (See appendix A for details).

Original: 5. S YNTHETIC D ATA E XPERIMENTS
Revised: 5. S YNTHETIC D ATA E XPERIMENTS

Original: Mode collapse and the quality of generations can be explicitly evaluated on synthetic data since the true distribution is well-deﬁned.
Revised: Mode collapse and the quality of generations can be explicitly evaluated on synthetic data since the true distribution is well-deﬁned.

Original: In this section, we evaluate the performance of the methods on[Figure] [Table] mixtures of Gaussian of known mode locations and distribution (See appendix B for details).
Revised: In this section, we evaluate the performance of the methods on mixtures of Gaussian of known mode locations and distribution (See appendix B for details).

Original: We use the same architecture for all the models, which is the same one used by Metz et al.
Revised: We use the same architecture for all the models, which is the same one used by Metz et al.

Original: (2017) and
Revised: (2017) and

Original: Srivastava et al.
Revised: Srivastava et al.

Original: Fig.
Revised: Fig.

Original: 2 displays the effect of each method on the 2D Ring and Grid data.
Revised: 2 displays the effect of each method on the 2D Ring and Grid data.

Original: As shown by the vanilla-GAN in the 2D Ring example (Fig. 2a), it can generate the highest quality  samples however it only captures a single mode.
Revised: As shown by the vanilla-GAN in the 2D Ring example (Fig. 2a), it can generate the highest quality [Figure] [Table] samples however it only captures a single mode.

Original: On the other extreme, the WGAN-GP on the 2D grid (Fig. 2k) captures almost all the modes in the true distribution, but this is simply because it generates highly scattered samples that do not precisely depict the true distribution.
Revised: On the other extreme, the WGAN-GP on the 2D grid (Fig. 2k) captures almost all the modes in the true distribution, but this is simply because it generates highly scattered samples that do not precisely depict the true distribution.

Original: GDPP-GAN (Fig.
Revised: GDPP-GAN (Fig.

Original: 2f,l) creates a precise representation of the true data distribution reﬂecting that the method learned an accurate structure manifold.
Revised: 2f,l) creates a precise representation of the true data distribution reﬂecting that the method learned an accurate structure manifold.

Original: Performance Evaluation:
Revised: Performance Evaluation:

Original: In every iteration of our experiments, we sample fake points from the generator and real points from the given distribution.
Revised: In every iteration of our experiments, we sample fake points from the generator and real points from the given distribution.

Original: The mode collapse is quantiﬁed by the number of real modes recovered in fake data, and the generation quality is quantiﬁed by the % of High Quality Samples.
Revised: The mode collapse is quantiﬁed by the number of real modes recovered in fake data, and the generation quality is quantiﬁed by the % of HighQuality Samples.

Original: A generated sample is counted as high-quality if it was sampled within three standard deviations in the case of the 2D Ring or 2D Grid, and ten standard deviations in the case of the 1200D data.
Revised: A generated sample is counted as high-quality if it was sampled within three standard deviations in the case of the 2D Ring or 2D Grid, and ten standard deviations in the case of the 1200D data.

Original: We train all the models for 25K iterations, except for VEEGAN which is trained for 100K iterations.
Revised: We train all the models for 25K iterations, except for VEEGAN which is trained for 100K iterations.

Original: At inference time, we generate 2500 samples from each of the trained models and measure both metrics.
Revised: At inference time, we generate 2500 samples from each of the trained models and measure both metrics.

Original: We report the numbers averaged over ﬁve runs with different random initialization in Table 1.
Revised: We report the numbers averaged over ﬁve runs with different random initialization in Table 1.

Original: GDPP-GAN clearly outperforms all other methods, for instance on the most challenging 1200D dataset that was designed to mimic a natural data distribution, bringing a 63% relative improvement in high quality samples and 15% in mode detection over its best competitor
Revised: GDPP-GAN clearly outperforms all other methods, for instance on the most challenging 1200D dataset that was designed to mimic a natural data distribution, bringing a 63% relative improvement in high-quality samples and 15% in mode detection over its best competitor

Original: WGAN-GP.
Revised: WGAN-GP.

Original: Ablation Study: We run a study on the 2D Ring and Grid data to show the individual effects of each component in our loss.
Revised: Ablation Study: We run a study on the 2D Ring and Grid data to show the individual effects of each component in our loss.

Original: As shown in Table 2, learningonly the  structure is prone tobeing affected by outlier structures introduced by the noise in the data and in the learning process.
Revised: As shown in Table 2, learning the unnormalized structure is prone to outlier structures introduced by the noise in the data and in the learning process.

Original: However, when scaling the structure loss by the true-data eigenvalues seems to better isolate the model from the noise that exist within the true-data features and only focus on learning the dominant structure.
Revised: However, when scaling the structure loss by the true-data eigenvalues seems to better disengage the model from the noise that exists within the true-data features and only focus on learning the dominant structure.

Original: Data-Efﬁciency: We evaluate the amount of training data needed by each method to reach the same local optima as evaluated by our two metrics on both the 2D Ring and Grid data.
Revised: Data-Efﬁciency: We evaluate the amount of training data needed by each method to reach the same local optima as evaluated by our two metrics on both the 2D Ring and Grid data.

Original: Since we are sampling the true-data from a mixture of Gaussians, we can generate an inﬁnite size of training data.
Revised: Since we are sampling the true-data from a mixture of Gaussians, we can generate an inﬁnite size of training data.

Original: Therefore, we can quantify the amount of the training data by using the batch-size while ﬁxing the number of back-propagation steps.
Revised: Therefore, we can quantify the amount of the training data by using the batch-size while ﬁxing the number of back-propagation steps.

Original: In this experiment (Fig. 3a), we run all the methods for the same number of iterations (25,000) and vary the batch size.
Revised: In this experiment (Fig. 3a), we run all the methods for the same number of iterations (25,000) and vary the batch size.

Original: However, WGAN-GP tends to capture higher quality samples with less data.
Revised: However, WGAN-GP tends to capture higher quality samples with fewer data.

Original: In the case of the 2D Grid data, GDPP-GAN performs on par with other methods for small amounts of data, yet it tends to signiﬁcantly outperform other methods on the quality of generated samples once trained on enough data.[Table] [Figure] [Figure]
Revised: In the case of the 2D Grid data, GDPP-GAN performs on par with other methods for small amounts of data, yet it tends to signiﬁcantly outperform other methods on the quality of generated samples once trained on enough data.

Original: Time-Efﬁciency: Another metric of interest is which method converges faster given the same amount of training data.
Revised: Time-Efﬁciency: Another property of interest is which method converges faster given the same amount of training data.

Original: In this experiment, we ﬁx the batch size on 512 and run the models for  variable number of iterations(Fig. 3b).
Revised: In this experiment, we ﬁx the batch size at 512 and train the models for a variable number of iterations (Fig. 3b).

Original: For the 2D Ring,  VEE-GAN captures  higher number of modes before GDPP-GAN, however they are of much lower quality than the ones generated by
Revised: For the 2D Ring, Only VEE-GAN captures a higher number of modes before GDPP-GAN, however, they are of much lower quality than the ones generated by

Original: GDPP-GAN.
Revised: GDPP-GAN.

Original: In the 2D Grid data, GDPP-GAN performs on par with unrolled-GAN for the ﬁrst 5,000 iterations, however it outperforms all the methods signiﬁcantly afterwards on both of the number of captured modes and the quality of generated samples.
Revised: In the 2D Grid data, GDPP-GAN performs on par with unrolled-GAN for the ﬁrst 5,000 iterations while the others are falling behind. After that our method signiﬁcantly outperforms [Table] Only diversity magnitude: ( L m )

Revised: [Table]

Original: 5. I MAGE GENERATION EXPERIMENTS
Revised: Only diversity structure: ( L s ) [Table] GDPP with unnormalized [Table]

Revised: [Table]

Revised: L m

Revised: [Table]

Revised: us )

Revised: [Table]

Revised: [Figure]

Revised: [Figure]

Revised: all the methods with respect to both the number of captured modes and the quality of generated samples.

Revised: 5. I MAGE GENERATION EXPERIMENTS

Original: We use the architecture described in (Gulrajani et al., 2017) for evaluating all the models on the
Revised: We use the architecture described in (Gulrajani et al., 2017) for evaluating all the models on the

Original: Stacked MNIST (Metz et al., 2017) and CIFAR10 datasets.
Revised: Stacked MNIST (Metz et al., 2017), CIFAR10, CelebA datasets.

Original: In the evaluation, we mainly focus on comparing to methods that adopt a change in the original adversarial loss.
Revised: In the evaluation, we mainly focus on comparing to methods that adopt a change in the original adversarial loss.

Original: Nevertheless, many of them can be deemed orthogonal to our contribution and can enhance the generation if embedded with our approach.
Revised: Nevertheless, many of them can be deemed orthogonal to our contribution and can enhance the generation if embedded with our approach.

Original: Since our method is cost-free, we observe that the training of GDPP-GAN has an indistinguishable running time than the DCGAN, which are the fastest models to ﬁnish training.
Revised: Since our method is cost-free, we observe that the training of GDPP-GAN has an indistinguishable running time than the DCGAN, which are the fastest models to train.

Revised: We also show that our loss is generic to any generative model by comparing it with the Variational AutoEncoder (VAE) Kingma & Welling (2013).

Original: Stacked-MNIST
Revised: Stacked-MNIST

Original: A variant of MNIST (LeCun, 1998) designed to increase the number of discrete modes in the data.
Revised: A variant of MNIST (LeCun, 1998) designed to increase the number of discrete modes in the data.

Original: The data is synthesized by stacking three randomly sampled MNIST digits along the color channel resulting in a 28x28x3 image.
Revised: The data is synthesized by stacking three randomly sampled MNIST digits along the color channel resulting in a 28x28x3 image.

Original: In this case, Stacked MNIST has 1000 discrete modes corresponding to the number of possible triplets of digits.
Revised: In this case, Stacked MNIST has 1000 discrete modes corresponding to the number of possible triplets of digits.

Original: Following (Gulrajani et al., 2017), we generate 50,000 images that are later used to train the networks.
Revised: Following (Gulrajani et al., 2017), we generate 50,000 images that are later used to train the networks.

Original: We train all the models for 15,000 iterations, except for DCGAN and unrolled-GAN that need 30,000 iterations to converge to a reasonable local-optima.
Revised: We train all the models for 15,000 iterations, except for DCGAN and unrolled-GAN that need 30,000 iterations to converge to a reasonable local-optima.

Original: We follow (Srivastava et al., 2017) to evaluate the methods on the number of recovered modes and the divergence between the true and fake distributions.
Revised: We follow (Srivastava et al., 2017) to evaluate the methods on the number of recovered modes and the divergence between the true and fake distributions.

Original: We samplefake images for all the models.
Revised: We sample 26000 fake images for all the models.

Original: We identify the mode of each generated image by using the classiﬁer mentioned in (Che et al., 2017) that is trained on  standard MNIST dataset to classify each channel of the fake sample.
Revised: We identify the mode of each generated image by using the classiﬁer mentioned in (Che et al., 2017) that is trained on the standard MNIST dataset to classify each channel of the fake sample.

Original: The quality of samples is evaluated by computing the KL-divergence between the generated label distribution and the training labels distribution.
Revised: The quality of samples is evaluated by computing the KL-divergence between the generated label distribution and the training labels distribution.

Revised: GDPP-GAN captures all the modes and generates a fake distribution that has the lowest KL-Divergence with the true-data.

Revised: Moreover, when applied on the VAE, it doubles the number of modes captured and cuts the KL-Divergence to half.

Original: CIFAR-10 Finally, we evaluate the methods on CIFAR-10 after training all the models for 100K iterations.
Revised: CIFAR-10 Finally, we evaluate the methods on CIFAR-10 after training all the models for 100K iterations.

Original: Unlike Stacked-MNIST, the modes are intractable in this dataset.
Revised: Unlike Stacked-MNIST, the modes are intractable in this dataset.

Original: To assess the performance on real-world images, we use two metrics: Inception Score for the generation quality and Inference-via-Optimization for diversity.
Revised: To assess the performance on real-world images, we use two metrics: Inception Score for the generation quality and Inference-via-Optimization for diversity.

Original: As shown in the Quantitative results on CIFAR and
Revised: As shown in the Quantitative results on CIFAR and

Original: Stacked MNIST (Table 3), GDPP-GAN consistently outperforms all other methods in both mode collapse and developing higher quality samples.
Revised: Stacked MNIST (Table 3), GDPP-GAN consistently outperforms all other methods in both mode collapse and developing higher quality samples.

Revised: When applying the GDPP on the VAE, it reduces the IvO by 63%, however, we note that both the inception-scores are considerably low which is also observed by Shmelkov et al.

Revised: (2018) when applying the VAE on CIFAR-10.

Original:  Inference-via-optimization (Metz et al., 2017) has been used to assess the severity of mode collapse in any generator by providing a way to compare real images with the nearest generated image.
Revised: [Table] Inference-via-optimization (Metz et al., 2017), has been used to assess the severity of mode collapse in any generator by providing a metric to compare real images with the nearest generated image.

Original: In the case of mode collapse, there are some real images for which this distance is large.
Revised: In the case of mode collapse, there are some real images for which this distance is large.

Original: We sample a real[Table] image x from the test set of real data.
Revised: We measure this metric by sampling a real image x from the test set of real data.

Original: Then we optimize the (cid:96) 2 loss between x and generated image G ( z ) by modifying the noise vector z .
Revised: Then we optimize the (cid:96) 2 loss between x and generated image G ( z ) by modifying the noise vector z .

Original: If a method attains low MSE, then it can be assumed that this method captures more modes than ones that attain a higher MSE. 
Revised: If a method attains low MSE, then it can be assumed that this method captures more modes than ones that attain a higher MSE. Fig.

Revised: 4 presents some real images with their nearest optimized generations.

Original:  As demonstrated by (Srivastava et al., 2017), this metric can be fooled by producing blurry images out of the optimization.
Revised: Randomly generated sample images can be seen in Appendix C. As demonstrated by (Srivastava et al., 2017), this metric can be fooled by producing blurry images out of the optimization.

Original: That is why  inception score is necessary in this evaluation.
Revised: That is why the inception score is necessary for this evaluation.

Original: Inception score (Salimans et al., 2016) is widely used as a metric for assessing the quality of images.
Revised: Inception score (Salimans et al., 2016) is widely used as a metric for assessing the quality of images.

Original: It bases its validity from the premise that every realistic image should be recognizable, which means that the score distribution for it must be, meaning that it is ideally dominated by one class.
Revised: It bases its validity from the premise that every realistic image should be recognizable by a standard architecture(e.g., Inception Network). Ideally, meaning that the score distribution for it must be dominated by one class.

Original: Randomly generated sample images can be seen in the appendix, and Fig.

Original: 9 presents some test images with their nearest generations.

Original: We also assess the stability of the training, by calculating the inception score at different  while training on CIFAR-10 (Fig. 5).
Revised: We also assess the stability of the training, by calculating the inception score at different stages while training on CIFAR-10 (Fig. 5).

Original: Evidently, DCGAN has the most unstable training, while GDPP-GAN tends to generate high-quality images the earliest on training with a stable increase.
Revised: Evidently, DCGAN has the least stable training with a high variation. However, adding GDPP penalty term to the generator loss results in generating high-quality images the earliest on training with a stable increase.

Revised: CelebA Finally, to evaluate the performance of our loss on large-scale

Revised: Adversarial training, we train

Revised: Progressive-Growing

Revised: GANs (Karras et al., 2017).

Original: C ONCLUSION
Revised: [Table] We show the effect of embedding our loss in adversarial training by adding it to the WGAN-GP this time instead of DCGAN loss, which is as well orthogonal to our loss. We train the model for 40K iterations corresponding to 4 scales up to 64 × 64 results on CelebA dataset (Liu et al., 2018). We evaluate the performance using Sliced Wasserstein Distance (SWD) (Peyr´e et al., 2017).

Revised: A small Wasserstein distance indicates that the distribution of the patches is similar.

Revised: That entails that real and fake images appear similar in both appearance and variation at this spatial resolution.

Revised: Accordingly, SWD metric can evaluate the quality of images as well as the severity of mode-collapse on large-scale datasets such as CelebA. Table 4 shows the average and minimum SWD metric across the last 10K training iterations.

Revised: For qualitative examples, refer to

Original:  In this work, we introduce a novel criterion to train generative networks on capturing a similar diversity tothe one of the true data by utilizing Determinantal Point Process(DPP) .
Revised: Fig. 12 in Appendix C.  C ONCLUSION In this work, we introduce a novel criterion to train generative networks on capturing a similar diversity to one of the true data by utilizing Determinantal Point Process(DPP).

Original: We apply our criterion to Generative Adversarial training  by learning a kernel via features extracted from the discriminator.
Revised: We apply our criterion to Generative Adversarial training and the Variational Autoencoder by learning a kernel via features extracted from the discriminator/encoder.

Original: We train the generator on optimizing a loss between the fake and real eigenvalues and eigenvectors of this kernel to simulate the diversity of the real data.
Revised: We train the generator on optimizing a loss between the fake and real, eigenvalues and eigenvectors of this kernel to simulate the diversity of the real data.

Original: Our GDPP framework accumulates many desirable properties: it does not require any extra trainable parameters, it works in an unsupervised setting, and outperforms state-of-the-art on synthetic data and real image datasets for both generation quality and invariance to mode collapse.
Revised: Our GDPP framework accumulates many desirable properties: it does not require any extra trainable parameters, it works in an unsupervised setting, and outperforms state-of-the-art on synthetic data and real image datasets for both generation quality and invariance to mode collapse.

Original: Furthermore, GDPP-GANs exhibit a stabilized  trainingas compared to other paradigms, and has been shown to be time and data efﬁcient as compared to stateof-the-art approaches.
Revised: Furthermore, GDPP-GANs exhibit a stabilized adversarial training and has been shown to be time and data efﬁcient as compared to state-of-the-art approaches.

Original: Moreover, the GDPP criterion is architecture and model invariant. Therefore, we aim on future work to explore incorporating the GDPP criterion with orthogonal methods such as adversarial feature learning, conditional GANs, and other generative models as well as deeper architectures.
Revised: Moreover, the GDPP criterion is architecture and model invariant, allowing it to be embedded with any variants of generative models such as adversarial feature learning or conditional GANs.

Original: S UPPLEMENTARY M ATERIAL A N ETWORK A RCHITECTURES
Revised: S UPPLEMENTARY M ATERIAL A N ETWORK A RCHITECTURES

Original: The architectures of the generator and discriminator networks employed in our experiments are given in Figure 6.
Revised: The architectures of the generator and discriminator networks employed in our experiments are given in Figure 6.

Revised: FC

Original: [Figure]
Revised: [Figure]

Revised: Z

Revised: Generator

Revised: FCTanh

Revised: Tanh

Revised: FC-

Revised: [Figure]

Revised: (x,y)

Revised: [Figure]

Revised: [Figure]

Revised: false) linear

Revised: Discriminator

Revised: FCBatchNorm

Revised: Relu

Revised: Reshape

Revised: Deconv2dBatchNorm

Revised: Relu

Revised: Deconv2dBatchNorm

Revised: Relu

Revised: Deconv2d-28*28*

Revised: Sigmoid

Revised: Z

Revised: Generator

Revised: Conv2DBatchNorm

Revised: L-Relu

Revised: Reshape

Revised: Conv2DBatchNorm

Revised: L-Relu

Revised: Conv2DBatchNorm

Revised: L

Revised: [Figure]

Revised: Relu

Revised: FCSigmoid

Revised: Image

Revised: [Figure]

Revised: (true, false)

Revised: Discriminator

Revised: [Figure]

Original: A. H YPERPARAMETERS
Revised: A. H YPERPARAMETERS

Original: In all of our experiments, we use Adam Optimizer with β 1 = 0 .
Revised: In all of our experiments, we use Adam Optimizer with β 1 = 0 .

Original: 5 and (cid:15) = 1 × 10 − 8 .
Revised: 5 and (cid:15) = 1 × 10 − 8 .

Original: For the synthetic data experiments, we follow the conﬁgurations used by VEEGAN and unrolled-GAN.
Revised: For the synthetic data experiments, we follow the conﬁgurations used by (Srivastava et al., 2017) and (Metz et al., 2017).

Original: We use 1 × 10 − 4 for the discriminator learning rate, and 1 × 10 − 3 for the generator learning rate.
Revised: We use 1 × 10 − 4 for the discriminator learning rate, and 1 × 10 − 3 for the generator learning rate.

Revised: For synthetic data we use a batch size of 512.

Original: For  the Stacked MNIST and CIFAR-10 dataset, we use 2 × 10 − 4 as the learning rate for both of the generator and the discriminator.
Revised: For Stacked-MNIST and CIFAR-10 we use a batch size of 64. For CIFAR-10, we use a batch size of 16. For the Stacked MNIST, CIFAR-10 and CelebA datasets, we use 2 × 10 − 4 as the learning rate for both of the generator and the discriminator.

Original: To  stabilize the training of DCGAN, we follow the protocol in WGAN-GP to train it by applying a learning rate scheduler.
Revised: To relatively stabilize the training of DCGAN, we follow the protocol in (Gulrajani et al., 2017) to train it by applying a learning rate scheduler.

Original: The decay is to happen with a ratio of 1 / (# max − iters ) at every iteration.
Revised: The decay is to happen with a ratio of 1 / (# max − iters ) at every iteration.

Original: B S YNTHETIC D ATA C OLLECTIONS
Revised: B S YNTHETIC D ATA C OLLECTIONS

Original: The ﬁrst data collection is introduced in (Metz et al., 2017) as a mixture of eight 2D Gaussian distributions arranged in a ring.
Revised: The ﬁrst data collection is introduced in (Metz et al., 2017) as a mixture of eight 2D Gaussian distributions arranged in a ring.

Original: This distribution is the easiest to mimic since it only requires the generated data to have an equal repulsion from the center of the distribution, even if it is not targeted to the modes.
Revised: This distribution is the easiest to mimic since it only requires the generated data to have an equal repulsion from the center of the distribution, even if it is not targeted to the modes.

Original: The second and third collections were introduced by (Srivastava et al., 2017).
Revised: The second and third collections were introduced by (Srivastava et al., 2017).

Original: In the second collection, there is a mixture of twenty-ﬁve 2D Gaussian distributions arranged in a grid.
Revised: In the second collection, there is a mixture of twenty-ﬁve 2D Gaussian distributions arranged in a grid.

Original: Unlike the ﬁrst collection, this one requires a more structured knowledge of the true data modes’ locations.
Revised: Unlike the ﬁrst collection, this one requires a more structured knowledge of the true data modes’ locations.

Original: The last collection is a mixture of ten 700 dimensional Gaussian distributions embedded in a 1200 dimensional space.
Revised: The last collection is a mixture of ten 700 dimensional Gaussian distributions embedded in a 1200 dimensional space.

Original: This mixture arrangement mimics the higher dimensional manifolds of natural images, and demonstrates the effectiveness of each method on manipulating sparse patterns.
Revised: This mixture arrangement mimics the higher dimensional manifolds of natural images, and demonstrates the effectiveness of each method on manipulating sparse patterns.

Original: C A DDITIONAL RESULTS
Revised: C A DDITIONAL RESULTS

Revised: other extreme, WGAN-GP tends to map the generated data to a disperse distribution covering all the modes with lower quality samples.