Title: IMAGE QUALITY ASSESSMENT TECHNIQUES IMPROVE TRAINING AND EVALUATION OF ENERGY-BASED GENERATIVE ADVERSARIAL NETWORKS

Abstract: We propose a new, multi-component energy function for energy-based Generative Adversarial Networks (GANs) based on methods from the image quality assessment literature. Our approach expands on the Boundary Equilibrium Generative Adversarial Network (BEGAN) by outlining some of the short-comings of the original energy and loss functions. We address these short-comings by incorporating an l 1 score, the Gradient Magnitude Similarity score, and a chrominance score into the new energy function. We then provide a set of systematic experiments that explore its hyper-parameters. We show that each of the energy function's components is able to represent a slightly different set of features, which require their own evaluation criteria to assess whether they have been adequately learned. We show that models using the new energy function are able to produce better image representations than the BEGAN model in predicted ways.

1. INTRODUCTION:

1.1 IMPROVING LEARNED REPRESENTATIONS FOR GENERATIVE MODELING Radford et al. (2015) demonstrated that Generative Adversarial Networks (GANs) are a good unsupervised technique for learning representations of images for the generative modeling of 2D images. Since then, a number of improvements have been made. First, Zhao et al. (2016) modified the error signal of the deep neural network from the original, single parameter criterion to a multi-parameter criterion using auto-encoder reconstruction loss. Berthelot et al. (2017) then further modified the loss function from a hinge loss to the Wasserstein distance between loss distributions. For each modification, the proposed changes improved the resulting output to visual inspection (see Appendix A Figure 4 , Row 1 for the output of the most recent, BEGAN model). We propose a new loss function, building on the changes of the BEGAN model (called the scaled BEGAN GMSM) that further modifies the loss function to handle a broader range of image features within its internal representation.

1.2. GENERATIVE ADVERSARIAL NETWORKS:

Generative Adversarial Networks are a form of two-sample or hypothesis testing that uses a classifier, called a discriminator, to distinguish between observed (training) data and data generated by the model or generator. Training is then simplified to a competing (i.e., adversarial) objective between the discriminator and generator, where the discriminator is trained to better differentiate training from generated data, and the generator is trained to better trick the discriminator into thinking its generated data is real. The convergence of a GAN is achieved when the generator and discriminator reach a Nash equilibrium, from a game theory point of view (Zhao et al., 2016) .

In the original GAN specification, the task is to learn the generator's distribution p G over data x (Goodfellow et al., 2014) . To accomplish this, one defines a generator function G(z; θ G ), which produces an image using a noise vector z as input, and G is a differentiable function with parameters θ G . The discriminator is then specified as a second function D(x; θ D ) that outputs a scalar representing the probability that x came from the data rather than p G . D is then trained to maximize the probability of assigning the correct labels to the data and the image output of G while G is trained to minimize the probability that D assigns its output to the fake class, or 1 -D(G(z)). Although G and D can be any differentiable functions, we will only consider deep convolutional neural networks in what follows. Zhao et al. (2016) initially proposed a shift from the original single-dimensional criterion-the scalar class probability-to a multidimensional criterion by constructing D as an autoencoder. The image output by the autoencoder can then be directly compared to the output of G using one of the many standard distance functions (e.g., l 1 norm, mean square error). However, Zhao et al. (2016) also proposed a new interpretation of the underlying GAN architecture in terms of an energy-based model (LeCun et al., 2006) .

1.3. ENERGY-BASED GENERATIVE ADVERSARIAL NETWORKS:

The basic idea of energy-based models (EBMs) is to map an input space to a single scalar or set of scalars (called its "energy") via the construction of a function (LeCun et al., 2006) . Learning in this framework modifies the energy surface such that desirable pairings get low energies while undesirable pairings get high energies. This framework allows for the interpretation of the discriminator (D) as an energy function that lacks any explicit probabilistic interpretation (Zhao et al., 2016) . In this view, the discriminator is a trainable cost function for the generator that assigns low energy values to regions of high data density and high energy to the opposite. The generator is then interpreted as a trainable parameterized function that produces samples in regions assigned low energy by the discriminator. To accomplish this setup, Zhao et al. (2016) first define the discriminator's energy function as the mean square error of the reconstruction loss of the autoencoder, or:

EQUATION

Zhao et al. ( 2016) then define the loss function for their discriminator using a form of margin loss.

EQUATION

where m is a constant and [•] + = max(0, •). They define the loss function for their generator:

EQUATION

The authors then prove that, if the system reaches a Nash equilibrium, then the generator will produce samples that cannot be distinguished from the dataset. Problematically, simple visual inspection can easily distinguish the generated images from the dataset.

1.4. DEFINING THE PROBLEM:

It is clear that, despite the mathematical proof of Zhao et al. (2016) , humans can distinguish the images generated by energy-based models from real images. There are two direct approaches that could provide insight into this problem, both of which are outlined in the original paper. The first approach that is discussed by Zhao et al. (2016) changes Equation 2to allow for better approximations than m. The BEGAN model takes this approach. The second approach addresses Equation 1, but was only implicitly addressed when (Zhao et al., 2016) chose to change the original GAN to use the reconstruction error of an autoencoder instead of a binary logistic energy function. We chose to take the latter approach while building on the work of BEGAN.

Our main contributions are as follows:

• An energy-based formulation of BEGAN's solution to the visual problem.

• An energy-based formulation of the problems with Equation 1.

• Experiments that explore the different hyper-parameters of the new energy function.

• Evaluations that provide greater detail into the learned representations of the model.

• A demonstration that scaled BEGAN+GMSM can be used to generate better quality images from the CelebA dataset at 128x128 pixel resolution than the original BEGAN model in quantifiable ways.

2. IMPROVING THE ENERGY-BASED MODEL OF GANS 2.1 BOUNDARY EQUILIBRIUM GENERATIVE ADVERSARIAL NETWORKS:

The Boundary Equilibrium Generative Adversarial Network (BEGAN) makes a number of modifications to the original energy-based approach. However, the most important contribution can be summarized in its changes to Equation 2. In place of the hinge loss, Berthelot et al. (2017) use the Wasserstein distance between the autoencoder reconstruction loss distributions of G and D. They also add three new hyper-parameters in place of m: k t , λ k , and γ. Using an energy-based approach, we get the following new equation:

EQUATION

The value of k t is then defined as:

k t+1 = k t + λ k (γE D (x) -E D (G(z))) for each t (5)

where

k t ∈ [0, 1] is the emphasis put on E(G(z)) at training step t for the gradient of E D , λ k is the learning rate for k, and γ ∈ [0, 1].

Both Equations 2 and 4 are describing the same phenomenon: the discriminator is doing well if either 1) it is properly reconstructing the real images or 2) it is detecting errors in the reconstruction of the generated images. Equation 4just changes how the model achieves that goal. In the original equation (Equation 2), we punish the discriminator (L D → ∞) when the generated input is doing well (E D (G(z)) → 0). In Equation 4, we reward the discriminator (L D → 0) when the generated input is doing poorly

(E D (G(z)) → ∞).

What is also different between Equations 2 and 4 is the way their boundaries function. In Equation 2, m only acts as a one directional boundary that removes the impact of the generated input on the discriminator if E D (G(z)) > m. In Equation 5, γE D (x) functions in a similar but more complex way by adding a dependency to E D (x). Instead of 2 conditions on either side of the boundary m, there are now four:

1. If γE D (x) > E D (G(z)) and E D (G(z)) → ∞, then L D → 0 and it is accelerating as k t → 1. 2. If γE D (x) > E D (G(z)) and E D (G(z)) → 0, then L D → E D (x) and it is accelerating as k t → 1. 3. If γE D (x) < E D (G(z)) and E D (G(z)) → ∞, then L D → 0 and it is decelerating as k t → 0. 4. If γE D (x) < E D (G(z)) and E D (G(z)) → 0, then L D → ∞ and it is decelerating as k t → 0.

The optimal condition is condition 1 Berthelot et al. (2017) . Thus, the BEGAN model tries to keep the energy of the generated output approaching the limit of the energy of the real images. As the latter will change over the course of learning, the resulting boundary dynamically establishes an equilibrium between the energy state of the real and generated input.1 

It is not particularly surprising that these modifications to Equation 2 show improvements. Zhao et al. (2016) devote an appendix section to the correct selection of m and explicitly mention that the "balance between... real and fake samples[s]" (italics theirs) is crucial to the correct selection of m. Unsurprisingly, a dynamically updated parameter that accounts for this balance is likely to be the best instantiation of the authors' intuitions and visual inspection of the resulting output supports this (see Berthelot et al., 2017) . We chose a slightly different approach to improving the proposed loss function by changing the original energy function (Equation 1).

2.2. FINDING A NEW ENERGY FUNCTION VIA IMAGE QUALITY ASSESSMENT:

In the original description of the energy-based approach to GANs, the energy function was defined as the mean square error (MSE) of the reconstruction loss of the autoencoder (Equation 1). Our first insight was a trivial generalization of Equation 1:

E(x) = δ(D(x), x) (6)

where δ is some distance function. This more general equation suggests that there are many possible distance functions that could be used to describe the reconstruction error and that the selection of δ is itself a design decision for the resulting energy and loss functions. Not surprisingly, an entire field of study exists that focuses on the construction of similar δ functions in the image domain: the field of image quality assessment (IQA).

The field of IQA focuses on evaluating the quality of digital images (Wang & Bovik, 2006) . IQA is a rich and diverse field that merits substantial further study. However, for the sake of this paper, we want to emphasize three important findings from this field. First, distance functions like δ are called full-reference IQA (or FR-IQA) functions because the reconstruction (D(x)) has a 'true' or undistorted reference image (x) which it can be evaluated from Wang et al. (2004) . Second, IQA researchers have known for a long time that MSE is a poor indicator of image quality (Wang & Bovik, 2006) . And third, there are numerous other functions that are better able to indicate image quality. We explain each of these points below.

One way to view the FR-IQA approach is in terms of a reference and distortion vector. In this view, an image is represented as a vector whose dimensions correspond with the pixels of the image. The reference image sets up the initial vector from the origin, which defines the original, perfect image. The distorted image is then defined as another vector defined from the origin. The vector that maps the reference image to the distorted image is called the distortion vector and FR-IQA studies how to evaluate different types of distortion vectors. In terms of our energy-based approach and Equation 6, the distortion vector is measured by δ and it defines the surface of the energy function.

MSE is one of the ways to measure distortion vectors. It is based in a paradigm that views the loss of quality in an image in terms of the visibility of an error signal, which MSE quantifies. Problematically, it has been shown that MSE actually only defines the length of a distortion vector not its type (Wang & Bovik, 2006) . For any given reference image vector, there are an entire hypersphere of other image vectors that can be reached by a distortion vector of a given size (i.e., that all have the same MSE from the reference image; see Figure 1 ). A number of different measurement techniques have been created that improve upon MSE (for a review, see Chandler, 2013). Often these techniques are defined in terms of the similarity (S) between the reference and distorted image, where δ = 1-S. One of the most notable improvements is the Structural Similarity Index (SSIM), which measures the similarity of the luminance, contrast, and structure of the reference and distorted image using the following similarity function:2 

S(v d , v r ) = 2v d v r + C v 2 d + v 2 r + C (7)

where v d is the distorted image vector, v r is the reference image vector, C is a constant, and all multiplications occur element-wise Wang & Bovik (2006). 3 This function has a number of desirable features. It is symmetric (i.e., S(v d , v r ) = S(v r , v d ), bounded by 1 (and 0 for x > 0), and it has a unique maximum of 1 only when v d = v r . Although we chose not to use SSIM as our energy function (δ) as it can only handle black-and-white images, its similarity function (Equation 7) informs our chosen technique.

The above discussion provides some insights into why visual inspection fails to show this correspondence between real and generated output of the resulting models, even though Zhao et al. (2016) proved that the generator should produce samples that cannot be distinguished from the dataset. The original proof by Zhao et al. (2016) did not account for Equation 1. Thus, when Zhao et al. (2016) show that their generated output should be indistinguishable from real images, what they are actually showing is that it should be indistinguishable from the real images plus some residual distortion vector described by δ. Yet, we have just shown that MSE (the author's chosen δ) can only constrain the length of the distortion vector, not its type. Consequently, it is entirely possible for two systems using MSE for δ to have both reached a Nash equilibrium, have the same energy distribution, and yet have radically different internal representations of the learned images. The energy function is as important as the loss function for defining the data distribution.

2.3. A NEW ENERGY FUNCTION:

Rather than assume that any one distance function would suffice to represent all of the various features of real images, we chose to use a multi-component approach for defining δ. In place of the luminance, contrast, and structural similarity of SSIM, we chose to evaluate the l 1 norm, the gradient magnitude similarity score (GMS), and a chrominance similarity score (Chrom). We outline the latter two in more detail below.

The GMS score and chrom scores derive from an FR-IQA model called the color Quality Score (cQS; Gupta et al., 2017) . The cQS uses GMS and chrom as its two components. First, it converts images to the YIQ color space model. In this model, the three channels correspond to the luminance information (Y) and the chrominance information (I and Q). Second, GMS is used to evaluate the local gradients across the reference and distorted images on the luminance dimension in order to compare their edges. This is performed by convolving a 3 × 3 Sobel filter in both the horizontal and vertical directions of each image to get the corresponding gradients. The horizontal and vertical gradients are then collapsed to the gradient magnitude of each image using the Euclidean distance. 4The similarity between the gradient magnitudes of the reference and distorted image are then compared using Equation 7. Third, Equation 7 is used to directly compute the similarity between the I and Q color dimensions of each image. The mean is then taken of the GMS score (resulting in the GMSM score) and the combined I and Q scores (resulting in the Chrom score).

In order to experimentally evaluate how each of the different components contribute to the underlying image representations, we defined the following, multi-component energy function:

EQUATION

where β d is the weight that determines the proportion of each δ to include for a given model, and D includes the l 1 norm, GMSM, and the chrominance part of cQS as individual δs. In what follows, we experimentally evaluate each of the energy function components(β) and some of their combinations.

3. EXPERIMENTS:



3.1. METHOD:

We conducted extensive quantitative and qualitative evaluation on the CelebA dataset of face images Liu et al. (2015) . This dataset has been used frequently in the past for evaluating GANs 

3.4. RESULTS:

GANs are used to generate different types of images. Which image components are important depends on the domain of these images. Our results suggest that models used in any particular GAN application should be customized to emphasize the relevant components-there is not a one-sizefits-all component choice. We discuss the results of our four evaluations below.

3.4.1. MEANS AND STANDARD DEVIATIONS OF ERROR SCORES:

Results were as expected: the three different distance functions captured different features of the underlying image representations. We compared all of the models in terms of their means and standard deviations of the error score of the associated distance functions (see Table 2 ). In particular, each of models 1-3 only used one of the distance functions and had the lowest error for the associated function (e.g., model 2 was trained with GMSM and has the lowest GMSM error score). Models 4-6 expanded on the first three models by examining the distance functions in different combinations. Model 5 (BEGAN+GMSM+Chrom) had the lowest chrominance error score and Model 6 (scaled BEGAN+GMSM) had the lowest scores for l 1 and GMSM of any model using a γ of 0.5.

For the models with γ set at 0.7, models 7-9 showed similar results to the previous scores. Model 8 (BEGAN+GMSM) scored the lowest GMSM score overall and model 9 (BEGAN+GMSM+Chrom) scored the lowest chrominance score of the models that did not suffer from modal collapse. For the two models that were trained to generate 128 × 128 pixel images, model 12 (scaled BE-GAN+GMSM) had the lowest error scores for l 1 and GMSM, and model 11 (BEGAN) had the lowest score for chrominance. Model 12 had the lowest l 1 score, overall.

3.4.2. VISUAL COMPARISON OF SIMILARITY SCORES:

Subjective visual comparison of the gradient magnitudes in column S of Figure 2 shows there are more black pixels for model 11 (row 11D) when comparing real images before and after autoencoding. This indicates a lower similarity in the autoencoder. Model 12 (row 12D) has a higher similarity between the original and autoencoded real images as indicated by fewer black pixels. This pattern continues for the generator output (rows 11G and 12G), but with greater similarity between the gradients of the original and autoencoded images than the real images (i.e., fewer black pixels overall).

The visual comparison of chrominance and related similarity score also weakly supported our hypotheses (see Figure 3 ). All of the models show a strong ability to capture the I dimension (blue-red) All images were normalized relative to their maximum value to increase luminance. Note that pink and purple approximate a similarity of 1, and green and blue approximate a similarity of 0 for I and Q dimensions, respectively. The increased gradient 'speckling' of model 12Q suggests an inverse relationship between the GMSM and chrominance distance functions.

4. OUTLOOK:

We bring an energy-based formulation to the BEGAN model and some of the problems of the energy function originally proposed in Zhao et al. (2016) . We proposed a new, multi-component energy function on the basis of research from the Image Quality Assessment literature. The scaled BEGAN+GMSM model produces better image representations than its competitors in ways that can be measured using subjective evaluations of the associated features (e.g., luminance gradient similarity, chrominance similarity). For future work, we would like to extend this research to encompass other datasets and FR-IQA energy functions. 

:

For a much more detailed and formal account that is beyond the scope of the current paper, see(Berthelot et al.,

2017).

The SSIM similarity function is reminiscent of the Dice-Sorensen distance function. It is worth noting that the Dice-Sorensen distance function does not satisfy the triangle inequality for setsGragera & Suppakitpaisarn (2016). Since sets are a restricted case for Equation7, where all the values are either 0 or 1, we can conclude that the corresponding distance of Equation 7 also fails to satisfy the triangle inequality. Consequently, it is not a true distance metric.

We use C = 0.0026 following the work on cQS described belowGupta et al. (2017).

For a detailed outline of the original GMS function, seeXue et al. (2014).

The code for the model and all related experiments are currently available on Github. Links will be included post-review.

Figures and Tables:

Figure 1: From left to right, the images are the original image, a contrast stretched image, an image with impulsive noise contamination, and a Gaussian smoothed image. Although these images differ greatly in quality, they all have the same MSE from the original image (about 400), suggesting that MSE is a limited technique for measuring image quality.

Figure 2: Comparison of the gradient (edges in the image) for models 11 (BEGAN) and 12 (scaled BEGAN+GMSM), where O is the original image, A is the autoencoded image, OG is the gradient of the original image, AG is the gradient of the autoencoded image, and S is the gradient magnitude similarity score for the discriminator (D) and generator (G). White equals greater similarity (better performance) and black equals lower similarity for the final column.

Figure 3: Comparison of the chrominance for models 9 (BEGAN+GMSM+Chrom), 11 (BEGAN) and 12 (scaled BEGAN+GMSM), where O is the original image, OC is the original image in the corresponding color space, A is the autoencoded image in the color space, and S is the chrominance similarity score. I and Q indicate the (blue-red) and (green-purple) color dimensions, respectively.All images were normalized relative to their maximum value to increase luminance. Note that pink and purple approximate a similarity of 1, and green and blue approximate a similarity of 0 for I and Q dimensions, respectively. The increased gradient 'speckling' of model 12Q suggests an inverse relationship between the GMSM and chrominance distance functions.

Figure 6: Measure of convergence and quality of the results for Models 11 (BEGAN; top images) and 12 (scaled BEGAN+GMSM; bottom images). The results were smoothed with a Gaussian with σ = 0.9. Images are displayed in 40,000 epoch increments starting with epoch 20,000 and going to 300,000. The output of earlier training epochs appear to be more youthful. As training proceeds, finer details are learned by the model, resulting in apparent increased age.

Lists the models, their discriminator mean error scores, and their standard deviations for the l 1 , GMSM, and chrominance distance functions over all training epochs. Bold values show the best scores for similar models. Double lines separate sets of similar models. Values that are both bold and italic indicate the best scores overall, excluding models that suffered from modal collapse. These results suggest that model training should be customized to emphasize the relevant components.

Table content: <table><tr><td/><td colspan="5">Discriminator Loss Statistics</td></tr><tr><td>Model #</td><td>l 1</td><td/><td colspan="2">GMSM</td><td colspan="2">Chrom</td></tr><tr><td/><td>M</td><td>σ</td><td>M</td><td>σ</td><td>M</td><td>σ</td></tr><tr><td>01</td><td colspan="6">0.12 0.02 0.24 0.02 0.68 0.08</td></tr><tr><td>02</td><td colspan="6">0.90 0.09 0.17 0.03 0.99 0.01</td></tr><tr><td>03</td><td colspan="6">0.51 0.02 0.52 0.04 0.46 0.08</td></tr><tr><td>04</td><td colspan="6">0.11 0.01 0.16 0.02 0.75 0.07</td></tr><tr><td>05</td><td colspan="6">0.13 0.02 0.20 0.02 0.41 0.05</td></tr><tr><td>06</td><td colspan="6">0.10 0.01 0.17 0.02 0.69 0.07</td></tr><tr><td>07</td><td colspan="6">0.10 0.01 0.22 0.01 0.63 0.08</td></tr><tr><td>08</td><td colspan="6">0.11 0.01 0.16 0.02 0.83 0.07</td></tr><tr><td>09</td><td colspan="6">0.13 0.02 0.20 0.02 0.42 0.06</td></tr><tr><td>10</td><td colspan="6">0.10 0.01 0.17 0.02 0.72 0.08</td></tr><tr><td>11</td><td colspan="6">0.09 0.01 0.29 0.01 0.58 0.08</td></tr><tr><td>12</td><td colspan="6">0.08 0.02 0.17 0.02 0.63 0.08</td></tr></table>

Peer review (iclr2018):
 This paper proposed some new energy function in the BEGAN (boundary equilibrium GAN framework), including l_1 score, Gradient magnitude similarity score, and chrominance score, which are motivated and borrowed from the image quality assessment techniques. These energy component in the objective function allows learning of different set of features and determination on whether the features are adequately represented. experiments on the using different hyper-parameters of the energy function, as well as visual inspections on the quality of the learned images, are presented. It appears to me that the novelty of the paper is limited, in that the main approach is built on the existing BEGAN framework with certain modifications. For example, the new energy function in equation (4) larges achieves similar goal as the original energy (1) proposed by Zhao et. al (2016), except that the margin loss in (1) is changed to a re-weighted linear loss, where the dynamic weighting scheme of k_t is borrowed from the work of Berthelot et. al (2017). It is not very clear why making such changes in the energy would supposedly make the results better, and no further discussions are provided. On the other hand, the several energy component introduced are simply choices of the similarity measures as motivated from the image quality assessment, and there are probably a lot more in the literature whose application can not be deemed as a significant contribution to either theories or algorithm designs in GAN. Many results from the experimental section rely on visual evaluations, such as in Figure~4 or 5; from these figures, it is difficult to clearly pick out the winning images. In Figure~5, for a fair evaluation on the performance of model interploations, the same human model should be used for competing methods, instead of applying different human models and different interpolation tasks in different methods.