F-Drop&Match: GANs With a Dead Zone in the High-Frequency ...

F-Drop&Match: GANs with a Dead Zone in the High-Frequency Domain

Shin’ya YamaguchiNTT

[email protected]

Sekitoshi KanaiNTT

[email protected]

Abstract

Generative adversarial networks built from deep convo-lutional neural networks (GANs) lack the ability to exactlyreplicate the high-frequency components of natural images.To alleviate this issue, we introduce two novel training tech-niques called frequency dropping (F-Drop) and frequencymatching (F-Match). The key idea of F-Drop is to filterout unnecessary high-frequency components from the in-put images of the discriminators. This simple modificationprevents the discriminators from being confused by pertur-bations of the high-frequency components. In addition, F-Drop makes the GANs focus on fitting in the low-frequencydomain, in which there are the dominant components ofnatural images. F-Match minimizes the difference betweenreal and fake images in the frequency domain for generatingmore realistic images. F-Match is implemented as a regu-larization term in the objective functions of the generators;it penalizes the batch mean error in the frequency domain.F-Match helps the generators to fit in the high-frequencydomain filtered out by F-Drop to the real image. We exper-imentally demonstrate that the combination of F-Drop andF-Match improves the generative performance of GANs inboth the frequency and spatial domain on multiple imagebenchmarks.

1. IntroductionGenerative adversarial networks built from deep convo-

lutional networks (GANs) [10, 11, 20, 22] have attractedmuch attention in the computer vision community andhave been utilized in various applications because they cansynthesize diverse images with high-fidelity to the targetdatasets. The training of GANs is formulated as a competi-tive game played by two neural networks called a generatorand a discriminator; the generator is optimized to producefake images that can fool the discriminator, and the discrim-inator is optimized to distinguish the real images from thefake images through min-max optimization. In theory, themodel replicates training data as the optimal result. How-ever, recent studies have revealed that GANs fail to replicate

StyleGAN2-ADA Ours

0.0

0.2

0.4

0.6

0.8

1.0

Figure 1. Sensitivities of discriminators in the frequency domain.The sensitivity is measured by single Fourier attack (SFA) [26],which perturbs each frequency component of an image. As thesensitivity, we plot the average differences between outputs of nor-mal and attacked discriminators over 128 images in the AFHQ-Catdataset (512 × 512) on each pixel. The differences in the low-frequency domain are located near the center of each figure, andthe differences in the high-frequency domains are at the edges.Our method outperforms the baseline (StyleGAN2-ADA) in termsof the robustness against the SFA on the high-frequency compo-nents.

data in the frequency domain [7, 8]. Durall et al. [7] andFrank et al. [8] have reported that the frequency character-istics of the generated images in the high-frequency domainare different from those of real images (we refer to this dif-ference as the frequency gap). They have also shown thatthe generated images can be easily detected as fakes with al-most 100% accuracy by assessing the frequency gap. Whilethe previous studies mainly focus on the aliasing causedby upsampling in CNNs as the cause of the frequency gap,modifying the upsampling is insufficient for correcting theflaws in the frequency domain [8]. In this study, we ex-plore another cause of the frequency gaps to reduce them.Since spatial and frequency domains are dual, reducing thefrequency gaps can improve the generative performances ofGANs in the spatial domain.

We hypothesize that the frequency gap is caused by thesensitivity of the discriminators to the perturbations in thehigh-frequency domain. In GANs for image generation,discriminators are usually implemented as CNN-based bi-nary classifiers. As shown in [26, 30], CNN-based classi-

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fiers are sensitive to perturbations of the frequency com-ponents. Moreover, Wang et al. [27] have reported thatCNN-based classifiers predict labels depending on high-frequency components that are hardly recognizable to hu-mans. Accordingly, we conjecture that the discriminators ofGANs are also sensitive to the high-frequency componentsof the input images. Indeed, our experiments demonstratethe sensitivity of the discriminator in the frequency domain:the output of the discriminator is significantly changed bysingle Fourier attack [26], which perturbs each frequencycomponent of an image (Fig. 1, left). The sensitivity of thediscriminators prevents the generators from learning databecause the generators are optimized to fool the discrimina-tors by perturbing high-frequency components rather thanby replicating data.

To alleviate the sensitivity of the discriminators and thefrequency gap, we present two novel techniques, calledfrequency dropping (F-Drop) and frequency matching (F-Match). The main idea of F-Drop is to filter out high-frequency components from the inputs of the discrimina-tors (for both real and generated images) and thereby thediscriminators concentrate on lower frequency components,which are the dominant components in natural images [29].We insert a low-pass filter, which filters out frequency com-ponents above a certain threshold from images, before theinput layer of the discriminators. F-Drop, (i) transformsRGB images into the frequency domain by using discretecosine transform (DCT), (ii) performs filtering in the fre-quency domain by element-wise multiplication, and (iii)transforms the images back into RGB space by using in-verse discrete cosine transform (IDCT). Since RGB im-ages are used as input, F-Drop does not require any mod-ifications to the original network architectures. By apply-ing F-Drop, the discriminators become robust against high-frequency perturbations (Fig. 1, right), and thus, the gen-erators can dedicate themselves to fooling the discrimina-tors by learning the remaining lower frequency components.However, since F-Drop simply transforms the input of thediscriminators, the generators are still free to synthesize thehigh-frequency components filtered out during the training.Hence, to synthesize realistic frequency components, wepropose F-Match, which minimizes the mean error in thefrequency domain. F-Match is a simple mini-batch-basedregularization term for the objective function of the genera-tors; it can utilize arbitrary frequency transformations (e.g.,DFT and DCT) and loss functions (e.g., the squared and ab-solute error). We experimentally found that the best func-tion for F-Match is the mean squared error in DCT space.Our experiments show that, in various settings, the com-bination of F-Drop and F-Match succeeds in synthesizingmore realistic images in both the frequency and spatial do-mains compared with the conventional techniques [4, 7, 8].Our contributions are summarized as follows:

• We demonstrate that the discriminators of GANs aresensitive to perturbations of high-frequency compo-nents through the experiments applying single Fourierattack to discriminators.

• We propose two simple techniques for GANs called F-Drop and F-Match for reducing the frequency gap be-tween real and generated images. F-Drop filters out thehigh-frequency components from the input images ofthe discriminators, and F-Match minimizes the meanerror in the frequency domain by adding a regulariza-tion term to the objective function of the generators.

• We confirm that our methods can improve the qualityof the generated images on various image datasets.

2. Related Work2.1. Frequency Gaps in Generative Models

Frequency gaps in GANs or CNN-based generative mod-els have been studied in recent papers [7, 8]. Durall etal. [7] and Frank et al. [8] have found that there are fre-quency gaps between real images and images generatedfrom CNN-based models by using discrete Fourier trans-form (DFT) and discrete cosine transform (DCT). Theyhave also found that the generated images are detected asfake by linear classifiers trained on the frequency compo-nents of the images. These studies have hypothesized thatupsampling in CNNs is a cause of the frequency gaps. Inparticular, Frank et al. [8] have shown that the frequencygaps can be reduced by modifying the upsampling in thegenerators (by using, e.g., binomial upsampling). However,the modification of upsampling is not sufficient for generat-ing undetectable fake images in the frequency domain. Fur-thermore, we empirically report that the modification maydegrade the generative performance of GANs in the spa-tial domain (Sec. H of the supplementary materials); thisdegradation has not been discussed in any previous works.In contrast to these works, we show that the discriminatorsof GANs are sensitive to high-frequency perturbations, andthat this sensitivity is also one of the causes of the frequencygaps.

For alleviating frequency gaps, Durall et al. [7] have pro-posed spectral regularization which minimizes the binarycross-entropy between the azimuthal integrals of the realand generated images in the frequency domain. Althoughspectral regularization has a similar form to F-Match, itminimizes the gaps between each generated image and themean value of the real images whereas F-Match minimizesthe gaps between the mean values of the generated and realimages over each mini-batch. Chen et al. [4] have proposeda similar technique called SSD, which modifies discrimi-nators by adding a classifier in the frequency domain andutilizes the output of the classifier to modulate the losses of

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the GANs. SSD does not use the gradients of the frequencyclassifier for training of GANs, whereas F-Match directlyuses the gradients of the loss in the frequency domain.

2.2. Sensitivity of CNNs for Frequency Components

In the context of adversarial attacks for CNN models,Tsuzuku and Sato [26] have pointed out a sensitivity ofCNNs in the frequency domain by conducting an analy-sis using their own black-box attack called single Fourierattack (SFA). SFA perturbs an image in the directions ofeach Fourier basis. Similar to [26], Yin et al. [30] haveshown that naturally trained CNNs are sensitive to high-frequency perturbations. In addition, Wang et al. [27] haveindicated that the output of CNN-based classifiers dependson the high-frequency components that are not visible tohumans. However, they have also shown that dropping thehigh-frequency components from the training does not de-grade the final test performances. Moreover, Xu et al. [29]have shown that dropping the high-frequency componentsfrom the input images of CNNs by thresholds helps to re-duce the input size and improves performance. In summary,the previous results provide two key insights: (i) CNNs-based classifiers have flaws in processing high-frequencycomponents in input images, and (ii) high-frequency com-ponents are not essential for training the classifiers. Theseinsights underlie the idea of F-Drop described in Sec. 5.1.

3. Background3.1. Generative Adversarial Networks

A generative adversarial network is composed of a gen-erator network Gθ parametrized by θ, and a discriminatornetwork Dφ parameterized by φ [10]. The Gθ generatesa fake sample xfake = Gθ(z) from a random noise z ∼ pz ,and theDφ distinguishes an observation x whether x comesfrom the data distribution pdata or not. The objective func-tions for training the discriminator and generator are

LDφ = −Ex∼pdatalogDφ(x)

−Ez∼pz log (1−Dφ(Gθ(z))), (1)LGθ = −Ez∼pz logDφ(Gθ(z)). (2)

By training ofGθ andDφ, Dφ learns to maximize the prob-ability of assigning a “real” label to real examples and a“fake” label to fake examples, whereas Gθ learns to maxi-mize the probability of Dφ’s failure of distinction. In the-ory, when Gθ and Dφ converge to the optimal point, thegenerator network Gθ implicitly replicates pdata.

In this paper, we mainly focus on GANs built fromCNNs. There are several variants, such as DCGAN [22],WGAN-GP [11], and SNGAN [20]. We can apply F-Dropand F-Match to any of these variants because they are de-signed as an additional masking layer in discriminators oras an additional regularization term.

3.2. Frequency Transformations

Here, we briefly summarize the foundations of discretecosine transform (DCT) that is used in F-Drop and F-Match.Note that, for simplicity, our discussion regards transforma-tions of a gray-scale square image X ∈ RH×H but it canbe easily extended to color images by performing the samecomputations on each channel.

Two-dimensional DCT [1, 9] is formulated as follows:

C(u, v) =2α(u)α(v)

H

H−1∑

i=0

H−1∑

j=0

X(i, j)c(i, j, u, v), (3)

where α(0) = 1/√

2, α(t) = 1 (for t 6= 0), and

c(i, j, u, v) = cos

[(2i+1)uπ

2H

]cos

[(2j+1)vπ

2H

].

where (i, j) represents a spatial pixel coordinate, (u, v) isa frequency coordinate. This form is called DCT-II. Wechoose DCT for F-Drop and F-Match as the default be-cause it does not have discontinuous boundaries that pro-duces high-frequency noise, in contrast to DFT [25]. As thetransformation from the frequency domain back to the spa-tial domain, we use two-dimensional inverse discrete cosinetransform (IDCT):

X(i, j) =2

H

H−1∑

u=0

H−1∑

v=0

α(u)α(v)C(u, v)c(i, j, u, v), (4)

where α(·) and c(·) as the same as in Eq. (3).

4. Analyzing GANs with Single Fourier AttackWe hypothesize that the frequency gap of GANs is

caused by the sensitivity of the discriminators to perturba-tions in the high-frequency domain. To confirm this hypoth-esis, we analyze GANs subjected to single Fourier attack(SFA) [26]. SFA attacks classification models by perturbingthe input images in the Fourier basis directions. For eachperturbation, SFA selects a single frequency component andcreates striped noise according to the selected component.A perturbation δ(u, v) of the frequency coordinate (u, v) foran H ×H image is defined as follows:

δ(u, v) =ε((1 + j)(FH)u ⊗ (FH)v

+ (1− j)(FH)H−u ⊗ (FH)H−v),(5)

where ε is a hyperparameter determining the size of the per-turbation, FH is the matrix of the Fourier basis and (FH)irepresents the i-th row of FH . Note that ⊗ means the Kro-necker product and j is the imaginary unit. We will useSFA to investigate the sensitivity of discriminators in thefrequency domain.

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xireal

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xifake

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D<latexit sha1_base64="nZgt/sIh7T9S0yaDYS4ZgQU6MOw=">AAACZHichVHLSsNAFD2Nr1qrrRZBEKRYFFflRgSLK0EXLq21D6hSkjhqaJqEJC1o8Qd0q7hwpSAifoYbf8BFf0AQlxXcuPA2DYiKeoeZOXPmnjtnZlTb0F2PqBWSenr7+gfCg5Gh6PBILD46VnCtuqOJvGYZllNSFVcYuinynu4ZomQ7Qqmphiiq1ZXOfrEhHFe3zE3vwBbbNWXP1Hd1TfGYyq5W4ilKkx/Jn0AOQApBrFvxG2xhBxY01FGDgAmPsQEFLrcyZBBs5rbRZM5hpPv7AkeIsLbOWYIzFGarPO7xqhywJq87NV1frfEpBneHlUnM0CPdUpse6I6e6f3XWk2/RsfLAc9qVyvsSux4Ivf2r6rGs4f9T9Wfnj3sIuN71dm77TOdW2hdfePwvJ1b2phpztIVvbD/S2rRPd/AbLxq11mxcYEIf4D8/bl/gsJ8Wqa0nF1ILWeCrwhjEtOY4/dexDLWsI48nytwglOchZ6kqJSQxrupUijQJPAlpKkPi0qJug==</latexit><latexit sha1_base64="nZgt/sIh7T9S0yaDYS4ZgQU6MOw=">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</latexit><latexit sha1_base64="nZgt/sIh7T9S0yaDYS4ZgQU6MOw=">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</latexit><latexit sha1_base64="nZgt/sIh7T9S0yaDYS4ZgQU6MOw=">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</latexit>

Predicting<latexit sha1_base64="INSwdxfCt9fP2L4rBiendIjwo+o=">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</latexit><latexit sha1_base64="INSwdxfCt9fP2L4rBiendIjwo+o=">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</latexit><latexit sha1_base64="INSwdxfCt9fP2L4rBiendIjwo+o=">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</latexit><latexit sha1_base64="INSwdxfCt9fP2L4rBiendIjwo+o=">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</latexit>

Real or Fake<latexit sha1_base64="67xOx9uFPzDRNmCr6TmRZcKZ2Bg=">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</latexit><latexit sha1_base64="67xOx9uFPzDRNmCr6TmRZcKZ2Bg=">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</latexit><latexit sha1_base64="67xOx9uFPzDRNmCr6TmRZcKZ2Bg=">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</latexit><latexit sha1_base64="67xOx9uFPzDRNmCr6TmRZcKZ2Bg=">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</latexit>

F-Match<latexit sha1_base64="ncvOBD32o5IZf0wMxtkEGvb2bsg=">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</latexit><latexit sha1_base64="ncvOBD32o5IZf0wMxtkEGvb2bsg=">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</latexit><latexit sha1_base64="ncvOBD32o5IZf0wMxtkEGvb2bsg=">AAACcXichVHLSsNAFD2N7/po1Y3iprQoglhuRFBcFQRxI/hqFbRKEqcamiYhmRZr8Qf8AQVXCiLiZ7jxB1z4CeKyghsX3qYB0aLeYWbOnLnnzpkZ3bVMXxI9R5S29o7Oru6eaG9f/0AsPjiU852yZ4is4ViOt61rvrBMW2SlKS2x7XpCK+mW2NKLi439rYrwfNOxN2XVFfmSdmibBdPQJFN7u1Icy9rS9IomjaPT/XiK0hREohWoIUghjFUnfotdHMCBgTJKELAhGVvQ4HPbgQqCy1weNeY8RmawL3CKKGvLnCU4Q2O2yOMhr3ZC1uZ1o6YfqA0+xeLusTKBcXqiO6rTI93TC338WqsW1Gh4qfKsN7XC3Y+djWy8/6sq8Sxx9KX607NEAfOBV5O9uwHTuIXR1FdOzusbC+vjtQm6plf2f0XP9MA3sCtvxs2aWL9ElD9A/fncrSA3k1Ypra7NpjLz4Vd0YwxJTPJ7zyGDZawiy+d6uMAVriN1ZVRJKMlmqhIJNcP4FsrUJwAfj3E=</latexit><latexit sha1_base64="ncvOBD32o5IZf0wMxtkEGvb2bsg=">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</latexit>

B<latexit sha1_base64="UCqE3P+xf/AAkah00fklFDHRyCI=">AAACZHichVHLSsNAFD2Nr1qrrRZBEKRYFFflRgSLK9GNS2vtA6qUJI4amiYhSQta/AHdKi5cKYiIn+HGH3DRHxDEZQU3LrxNA6Ki3mFmzpy5586ZGdU2dNcjaoWknt6+/oHwYGQoOjwSi4+OFVyr7mgir1mG5ZRUxRWGboq8p3uGKNmOUGqqIYpqdbWzX2wIx9Utc9M7sMV2Tdkz9V1dUzymsiuVeIrS5EfyJ5ADkEIQ61b8BlvYgQUNddQgYMJjbECBy60MGQSbuW00mXMY6f6+wBEirK1zluAMhdkqj3u8KgesyetOTddXa3yKwd1hZRIz9Ei31KYHuqNnev+1VtOv0fFywLPa1Qq7EjueyL39q6rx7GH/U/WnZw+7yPhedfZu+0znFlpX3zg8b+eWNmaas3RFL+z/klp0zzcwG6/adVZsXCDCHyB/f+6foDCfliktZxdSy5ngK8KYxDTm+L0XsYw1rCPP5wqc4BRnoScpKiWk8W6qFAo0CXwJaeoDh0qJuA==</latexit><latexit sha1_base64="UCqE3P+xf/AAkah00fklFDHRyCI=">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</latexit><latexit sha1_base64="UCqE3P+xf/AAkah00fklFDHRyCI=">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</latexit><latexit sha1_base64="UCqE3P+xf/AAkah00fklFDHRyCI=">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</latexit>

F(xireal)

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F(xifake)

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Figure 2. Illustration of proposed methods

As a preliminary experiment, we tested ResNet-basedSNGANs [20] trained on the CelebA dataset [19] (The de-tails of the training are shown in Sec. 6.1). For SFA, we setε to 10/255. The left-hand side of Figure 1 visualizes theresults of SFA. The visualization procedure followed [26].Each pixel coordinate corresponds to a frequency compo-nent used for SFA, and each pixel represents the absolutedifferences |D(x)−D(x+ δ(u, v))|, i.e., the sensitivity tothe perturbation. Note that the differences are normalized to[0, 1] by dividing with the maximum value of the SNGANsand ours. We can see that the SNGANs are sensitive tohigh frequency perturbations like the results in [30]. Thisindicates that the discriminators are easily fooled by per-turbing the high-frequency domain and their sensitivity inthis regard leads to the frequency gaps because the genera-tors focus on synthesizing the high-frequency perturbationsrather than a realistic image. Since the high-frequency com-ponents are not necessary for training CNNs, as discussedin Sec. 2.2, we will examine a method of filtering out themfrom the input images.

5. Proposed Method

Figure 2 illustrates the overview of F-Drop and F-Match.F-Drop filters out the high-frequency components from theinput images for discriminators, while F-Match is a regu-larization method for generators, which penalizes the mini-batch mean error in the frequency domain between the realand generated images. F-Drop and F-Match are indepen-dent of each other and can be easily incorporated in the ar-chitectures of GANs.

5.1. Frequency Dropping

First, we introduce the idea of frequency dropping (F-Drop). As discussed in Sec. 2.2 and 4, the discriminatorsof GANs are sensitive to the high-frequency componentsof the input images, but can be trained without the high-frequency components. F-Drop is based on these insights;it filters out the high-frequency components from images bymasking with a user-defined threshold parameter γ ∈ [0, 1].

The procedure of F-Drop is quite simple: (i) transform aninput image into the frequency domain, (ii) drop the high-frequency components, and (iii) transform the frequencycomponents back into the spatial domain. F-Drop trans-forms an input color image R3×H×W for the discriminatorsas follows:

Drop(x, γ) = F−1 (F(x)�M(γ)) , (6)

where F is a frequency transform function, such as DCT,F−1 is an inverse frequency transform function, such asIDCT, and M(γ) ∈ R3×H×W is a mask matrix for filter-ing specific frequency components. Note that � denoteselement-wise multiplication. We chose DCT to be F andIDCT to be F−1. An element in a coordinate (c, u, v) ofthe mask matrix M(γ) is defined as

M cu,v(γ) =

{1 (√u2 + v2 ≤ γ

√H2 +W 2)

0 (√u2 + v2 > γ

√H2 +W 2).

(7)

That is, we drop the high-frequency components of coor-dinates farther away than γ

√H2 +W 2, which is the Eu-

clidean distance from the origin point (0, 0) (i.e., the directcurrent component). After masking, we utilize the remain-ing lower frequency components for training the GANs(Fig. 2, top). We can adjust the cutoff frequency with thethreshold hyperparameter γ. As defined in Eq. (7), the chan-nels c share the mask element, and thus, the Drop(·) calcu-lation can be implemented by broadcasting a single chan-nel mask M(γ) ∈ RH×W . Since all of the operations inDrop(·) are differentiable with respect to the input data, wecan train the models by gradient descent via backpropaga-tion in an end-to-end fashion.

5.2. Frequency Matching

Frequency matching (F-Match) is for minimizing the fre-quency gap between the real and generated images. The keyidea is matching the frequency characteristics of the real andgenerated images. F-Match minimizes the frequency gapsby using the mini-batch statistics of the images because animage generated from GANs does not have a one-to-onecorrespondence to a real image. This regularization strat-egy is commonly used by methods such as feature matchingin [24] and MMD-GAN [18]. The loss function of F-Matchis formalized as follows:

Lmatch = d(Xreal, Xfake

), (8)

Xreal =1

B

B−1∑

i=0

F(xireal), Xfake =1

B

B−1∑

i=0

F(xifake),

where d(·) is an error function, B is batch size for eachtraining iteration, and xireal is the i-th real image, and xifakeis the i-th generated image from the GANs. d(·) and F(·)

6746

can be set to an arbitrary error function (e.g., squared error)and arbitrary frequency transform (e.g., DCT). In the sup-plementary materials, we evaluate various combinations ofd(·) and F(·) and show that the mean squared error (MSE)in DCT space is the best choice. We use the following MSE-based function:

dMSE =1

HW

H∑

u

W∑

v

(Xreal(u, v)− Xfake(u, v)

)2, (9)

where X(u, v) is the (u, v) coordinate of X . In the op-timization, Lmatch is added as a regularization term to theobjective function defined in Eq. (2):

LGθ = Ez∼pz logDφ(Gθ(z)) + λLmatch, (10)

where λ is a balancing hyperparameter. As discussed inSec. 2.1, spectral regularization (SR) [7] is defined in a sim-ilar form to F-Match. Following [7], the loss function of SRfor H ×H square images is defined as

LSR =1

B

B−1∑

i=0

dSR(Xreal,F(xifake)

), (11)

dSR =− 1

H/2− 1

H/2−1∑

r=0

BCE(A(Xreal, r), A(Xfake, r)

),

where BCE(·) is the binary cross entropy function andA(X, r) is the azimuthal integral 1

2π

∫ 2π

0|X(r, θ)|dθ,

which approximates 2D DFT images into 1D signals withrespect to the radial distance r in polar coordinates (r, θ).Note that SR differs from F-Match in that it uses a singlegenerated image for minimizing the frequency gaps.

The final objective functions using F-Drop and F-Matchare:

LDφ =− Ex∼pdatalogDφ(Drop(x, γ))

− Ez∼pz log (1−Dφ(Drop(Gθ(z), γ))),(12)

LGθ = −Ez∼pz logDφ(Drop(Gθ(z), γ))+λLmatch. (13)

The overall training procedure with F-Drop and F-Match issummarized in Algorithm 1. Note that, unlike the trainingof normal GANs, we pre-fetch the input real images {xi}for calculating the loss function of F-Match (Eq. 8) on line4. GetSample and GenNoise are functions for fetchingbatch images and for generating batch noise from a normaldistribution.

6. ExperimentsWe evaluate our proposed methods (F-Drop and F-

Match) by comparing them with naive baselines and the ex-isting methods [4, 7, 8]. We evaluate our methods in termsof (i) quantitative metrics for GANs (main evaluations),

Algorithm 1 Training of GAN with F-Drop and F-MatchRequire: Batchsize B, learning rate ηθ, ηφ, number of critics K, hyper-

parameters γ, λ1: Randomly initialize parameters θ, φ2: while not convergent do3: for k = 1 to K do4: {xireal}

B−1i=0 ← GetSample(B)

5: {zi}B−1i=0 ← GenNoise(B)

6: if k = 1 then7: {xifake}

B−1i=0 ← {Gθ(z

i)}B−1i=0

8: Lmatch ← d(

1B

∑B−1i=0 F(x

ireal),

1B

∑B−1i=0 F(x

ifake)

)9: LGθ ← −

∑Bi logDφ(Drop(xifake, γ))) + λLmatch

10: θ← θ − ηθ∇θLGθ11: end if12: LDφ ← −

∑B−1i=0 logDφ(Drop(xireal, γ))

13: −∑B−1i=0 log(1−Dφ(Drop(Gθ(zi), γ)))

14: φ← φ− ηφ∇φLDφ15: end for16: end while

(ii) sensitivity to the high-frequency components (frequencysensitivity analysis), (iii) the results in the state-of-the-artsettings using StyleGAN2-ADA, and (iv) the quality of thegenerated images. The supplementary materials contain anablation study on F-Match, an analysis of hyperparametersensitivity of F-Drop, and fake detection in the frequencydomain (fake detection).

6.1. Setup

Datasets We used the six different image datasets:CIFAR-10 and CIFAR-100 (32× 32) [15], TinyImageNet(32×32) [28], STL-10 (48×48) [6], CelebA (128×128) [19],and ImageNet (128×128) [23]. These datasets have beenused for testing the benchmarks of GANs [3, 11, 17, 20, 31].We applied center cropping and resizing to the images ofTinyImageNet, CelebA, and ImageNet before the training.For training, we normalized images into a range of [−1, 1].

GAN Baselines As a baseline, we chose spectral normal-ization GAN (SNGAN) with ResNet-backbone architec-tures [20]. As additional baselines, we tested Binomial [8],which replaces the bilinear upsampling filters in the gener-ators with the low-pass filters based on a binomial distri-bution, spectral regularization (SR) [7], which minimizesthe gaps of the azimuthal integral in DFT space by usingEq. (11) (see Sec. 5.2), and SSD-GAN [4], which adds afrequency classifier in DFT space (using the azimuthal inte-gral) to the discriminator and utilizes the output of the clas-sifier to modulate the loss functions of the GANs. We usedthe Binomial-5 kernel, following Frank et al. [8]. SR wasbased on one in the author’s public code repository and used

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Table 1. Mean frequency gaps between real and fake imagesCIFAR-10 CIFAR-100 TinyImageNet STL-10 CelebA ImageNet

SNGAN 6.89 7.01 9.83 4.19 4.49 4.83Binomial [8] 7.85 5.83 9.96 4.30 4.74 4.55SR [7] 6.12 6.80 9.77 3.98 4.48 5.70SSD-GAN [4] 6.39 6.80 9.97 4.59 4.47 4.80F-Drop 5.94 6.36 9.29 3.87 4.60 5.39F-Match 4.84 4.87 7.36 4.04 4.46 4.52F-Drop&Match 3.93 4.16 6.49 3.86 4.43 4.41

Figure 3. Comparison of average DCT coefficients (top: CIFAR-100, bottom: CelebA). The visualization protocols follow those ofFrank et al. [8].

λ as 1.0×10−5.1,2 For SSD-GAN, the implementation wascomposed of the author’s code, and we used λ = 0.5, fol-lowing [4].3 For the labeled datasets (i.e., CIFAR-10/-100,TinyImageNet, and ImageNet), we used conditional batchnormalization for generators and the projection discrimina-tor following [21]. We evaluate other representative GANvariants, including deep convolutional GAN (DCGAN) [22]and Wasserstein GAN with a gradient penalty (WGAN-GP) [11] instead of SNGAN in Sec. 6.2. We implementedthe architectures of GANs with the open-source repositoryof Lee et al. [16]. More details of the training and evaluationsettings of GANs appear in the supplementary materials.

6.2. Main Evaluations

Frequency Gaps First, we evaluate the reduction in thefrequency gaps. The following total absolute difference inDCT space was used as a measure of the frequency gap:

1

HW

H∑

u

W∑

v

∣∣Xreal(u, v)− Xfake(u, v)∣∣ , (14)

where X(u, v) is defined in Eq. (8). We computed thegaps between 10k real and generated images, where thereal images were randomly selected from each dataset. Ta-ble 1 lists the mean frequency gaps measured by Eq. (14).The F-Drop&Match column represents the performances ofSNGAN simultaneously applying F-Drop and F-Match. Vi-sualizations of the frequency characteristics are shown inFig. 3, where the pixels in the upper left represent lower

1https://github.com/cc-hpc-itwm/UpConv/2We used PyTorch to implement the differentiable azimuthal integral

because the author’s implementation, which uses Numpy, is not differen-tiable. More detailed discussions appear in the supplementary materials.

3https://github.com/cyq373/SSD-GAN

frequency components and ones in the lower right rep-resent higher frequency components. The figure and ta-ble show that F-Drop&Match significantly reduced the fre-quency gaps in all datasets and replicated more realistic fre-quency characteristics compared with the other methods. Ina few cases, F-Drop by itself did not reduce the gaps. This isbecause F-Drop allows the generators to synthesize the fil-tered out high-frequency components, and thus, the gener-ated images contain high-frequency components at random.On the other hand, F-Match by itself reduced the gaps in allcases, since it directly minimizes the frequency characteris-tics. In Fig. 3, the results of F-Match show frequency gapsin the middle range of the frequency domain more so thanthe results of F-Drop&Match. This is because the genera-tor of F-Match (by itself) focuses on high-frequency com-ponents because of the sensitivity of the discriminators tothe high-frequency domain. These results indicate that F-Drop&Match reduces the frequency gaps by complemen-tarily combining filtering and direct minimization. Further-more, F-Drop&Match outperformed the other frequency-oriented methods, i.e., Binomial, SR, and SSD-GAN. Thepoorer performance of these other methods is probably be-cause they do not take account of the sensitivity of the dis-criminators to the high-frequency domain. The sensitivityof the other methods is discussed in Sec. 6.3. Here, we dis-cuss other reasons why Binomial, SR, and SSD-GAN areinferior to our methods. In the case of Binomial, the bi-nomial upsampling suppresses the high-frequency compo-nents in the generators, but does not explicitly regularizesthe models to learn the frequency characteristics. More-over, we found that Binomial tends to degrade the genera-tive performance in the spatial domain (see the evaluationsdescribed below). For SR and SSD-GAN, the performancegains sensitively depend on the dataset. This behavior re-flects the 1D approximation with the azimuthal integral,which implicitly assumes that the real frequency character-istics are distributed in a concentric pattern in DFT space.Since SSD-GAN does not use the gradients from the fre-quency classifier for updating the GANs, its performancegains may be unstable. Meanwhile, F-Match directly mini-mizes the gaps for each frequency component in an end-to-end fashion and performs stably on the various datasets.

FID/KID/IS Second, we measured the Frechet inceptiondistance (FID) [12], kernel inception distance (KID) [2],and inception score (IS) [24]. We computed these mea-sures on 100k real and generated images for the 128×128datasets, and 50k real and generated images for the 32×32datasets and STL-10. Table 2, 3, and 4 show the scores ofFID/KID/IS for each combination of dataset, method, andGAN variant. Note that ↓means lower is better and ↑meanshigher is better. F-Drop by itself and F-Match by itself out-performed the baselines in many cases. More importantly,

6748

Table 2. Performance comparison on the 32× 32 datasetsCIFAR-10 CIFAR-100 TinyImageNet

FID (↓) KID×10−3 (↓) IS (↑) FID (↓) KID×10−3 (↓) IS (↑) FID (↓) KID×10−3 (↓) IS (↑)SNGAN 14.3±0.73 9.20±0.57 8.25±0.14 15.2±0.25 9.76±0.35 8.91±0.04 21.8±3.01 12.3±3.43 6.35±0.28

Binomial [8] 35.9±0.90 21.9±1.89 6.60±0.15 23.7±0.70 14.3±0.47 8.09±0.08 53.9±4.26 30.8±8.91 5.34±0.29

SR [7] 12.2±0.27 7.73±2.73 8.43±0.01 14.7±0.27 9.56±0.49 8.94±0.05 23.8±2.01 18.9±2.00 5.96±0.25

SSD-GAN [4] 13.4±0.13 8.72±0.21 8.32±0.11 14.9±0.88 9.32±0.78 9.01±0.31 21.1±1.51 12.9±1.72 6.50±0.23

F-Drop 14.1±0.81 9.11±0.21 8.31±0.18 15.1±0.15 9.47±0.29 8.93±0.05 20.4±0.46 11.5±1.18 6.49±0.08

F-Match 12.8±0.53 7.90±0.32 8.45±0.12 14.7±0.66 9.09±0.89 9.17±0.24 20.9±0.24 12.7±0.46 6.41±0.24

F-Drop&Match 10.7±0.92 7.15±0.58 8.45±0.06 13.8±0.34 8.99±0.49 9.16±0.00 18.9±1.08 10.3±0.52 6.55±0.14

Table 3. Performance comparison on larger image datasetsSTL-10 (48× 48) CelebA (128× 128) ImageNet (128× 128)

FID (↓) KID×10−3 (↓) IS (↑) FID (↓) KID×10−3 (↓) IS (↑) FID (↓) KID×10−3 (↓) IS (↑)SNGAN 34.7±1.26 32.0±0.91 8.68±0.08 7.98±0.13 4.45±0.42 3.02±0.07 62.5±1.16 63.5±0.80 14.1±0.34

Binomial [8] 34.9±0.44 32.4±1.01 8.66±0.09 37.9±6.57 22.3±0.95 2.86±0.02 76.6±6.93 74.1±6.15 11.6±1.14

SR [7] 38.1±0.74 34.9±0.87 8.49±0.02 11.2±0.74 5.67±0.87 2.91±0.06 64.0±1.52 64.9±2.35 13.9±0.49

SSD-GAN [4] 35.6±0.25 32.2±0.68 8.77±0.03 7.88±0.64 4.20±0.97 3.05±0.07 61.2±0.49 61.6±1.69 14.3±0.08

F-Drop 34.7±0.75 31.8±1.09 8.75±0.07 6.86±0.47 3.92±0.64 3.09±0.06 61.0±0.59 60.9±1.86 14.2±0.21

F-Match 34.0±0.72 31.1±0.76 8.79±0.05 6.78±0.16 3.73±0.18 3.08±0.04 62.0±1.33 62.2±1.35 14.4±0.18

F-Drop&Match 33.8±0.66 30.4±0.83 8.85±0.15 6.78±0.11 3.61±0.10 3.16±0.05 60.4±0.71 60.5±0.51 14.5±0.30

Table 4. Performance comparison on GAN variants (CIFAR-100)DCGAN [22] WGAN-GP [11]

FID (↓) KID×10−3 (↓) IS (↑) FID (↓) KID×10−3 (↓) IS (↑)Baseline 27.2±1.15 16.2±1.69 7.16±0.26 25.2±0.20 21.2±0.33 7.72±0.03

Binomial [8] 49.8±3.78 28.6±2.38 6.28±0.18 25.1±0.48 19.8±0.77 7.65±0.07

SR [7] 40.8±2.28 24.4±3.07 6.24±0.27 40.9±3.68 22.7±2.84 6.28±0.45

SSD-GAN [4] 34.2±1.58 18.6±1.62 6.53±0.23 45.0±1.76 30.1±1.96 6.04±0.32

F-Drop 25.9±0.45 15.8±0.27 7.15±0.05 23.8±0.28 19.3±0.52 7.85±0.01

F-Match 26.5±0.73 16.3±1.06 7.23±0.17 24.9±0.14 20.2±0.21 7.59±0.11

F-Drop&Match 25.2±1.17 15.4±0.44 7.45±0.07 23.9±0.53 18.9±0.79 7.97±0.02

F-Drop&Match performed the best in all cases. Binomialunderperformed the baseline in almost all cases. Similarto the evaluation of the frequency gaps, the performancesof SR and SSD-GAN sensitively depend on the datasets,while our methods stably outperform the baselines. Theseresults indicate that our methods can flexibly help GANs toreplicate the real images in both the frequency and spatialdomain.

6.3. Frequency Sensitivity Analysis

As shown in Sec. 4, the discriminators of GANs are sen-sitive to the perturbations in the high-frequency domain.We evaluate the sensitivity of our methods by conductingan SFA analysis. Figure 4 compares the baselines and ourmethods in terms of the results of SFA perturbing each fre-quency component on CelebA. Figure 5 shows the resultsof SFA on multiple datasets. We used the same visualiza-tion protocol as in Sec. 4. We also tested Binomial, SR,and SSD-GAN on CelebA (Fig. 4); the results on the otherdatasets appear in the supplementary materials and lead tothe same conclusions as given below. F-Drop&Match out-performed all of the baselines. Since the robust frequency

SNGAN Binomial SR SSD-GAN F-Drop F-Match F-Drop&Match

0.0

0.2

0.4

0.6

0.8

1.0

Figure 4. Sensitivity analysis by SFA [26] on CelebA

SNG

AN

CIFAR-10 CIFAR-100 TinyImageNet STL-10 ImageNet

0.0

0.2

0.4

0.6

0.8

1.0

F-D

rop&

Mat

ch

CIFAR-10 CIFAR-100 TinyImageNet STL-10 ImageNet

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5. Sensitivity analysis by SFA [26] on multiple datasets

domains of F-Drop&Match seem to be the union of the ro-bust frequency domains of F-Drop and ones of F-Match,we see that the robustness of F-Drop&Match comes fromcombining F-Drop and F-Match complementarily. Moreimportantly, in Fig. 4, we see that Binomial, SR, and SSD-GAN are not robust against the low to middle range of thefrequency domain. We consider that this is because thesemethods, unlike F-Drop, feed the whole input images in-cluding their high-frequency components into the discrim-inators, and thus, the discriminators have trouble focusingon the lower frequency domain. These results suggest thatthe discriminators of F-Drop&Match can focus on the lowerfrequency and the generators indirectly adjust their trainingto learn realistic frequency components by combining F-Drop and F-Match complementarily.

6749

Table 5. Evaluation on StyleGAN2-ADA (FID)FFHQ Cat Dog Wild

(256× 256) (512× 512)

StyleGAN2-ADA [13] 4.30 3.55 7.40 3.05F-Drop&Match 4.05 3.36 7.21 2.62

Rea

lSt

yleG

AN

2-A

DA

Our

s

Figure 6. Visualization of real and generated images (AFHQ-Cat)

6.4. Evaluation on StyleGAN2-ADA

Here, we show extra results evaluated on state-of-theart settings. We chose StyleGAN2-ADA [13] as the base-line and used the implementation provided by the authors.4

We evaluated models on the four high-resolution datasets:FFHQ [14], and AFHQ-Dog/Cat/Wild [5]. The training set-tings of F-Drop&Match are shared with that of the previ-ous work [13]; we used paper256 setting for FFHQ andpaper512 setting for the AFHQ datasets, which are presetin the repository. The hyperparameters of λwere 1.0×10−7

for FFHQ and 1.0 × 10−8 for the AFHQ datasets. Table 5summarizes the results of FID. Our method succeeded toimprove the baseline. As the same trend in Section 6.3,Figure 1 shows that our method can prevent the discrim-inators to be fooled by the high-frequency components ininput. Thus, F-Drop&Match can work well even on high-resolution datasets with state-of-the-art GAN variant.

6.5. Qualitative Results

Lastly, we provide visualizations of the generated im-ages. Figure 6 illustrates the generated images from theStyleGAN2 and F-Drop&Match (ours) models trained onthe AFHQ-Cat dataset; the training settings are shared withSection 6.4. The generated images are randomly selected.We emphasize again that we did not use M(γ) of F-Dropin the evaluation after training. We can see that both

4https://github.com/NVlabs/stylegan2-ada-pytorch

StyleGAN2-ADA and ours synthesized rough shapes of catfaces that are composed of lower frequency components.Meanwhile, ours was superior to StyleGAN2-ADA at syn-thesizing more detailed information such as hair which iscomposed of higher frequency components, while keepingthe information on the lower frequency components suchas the positions of facial parts. These results indicate thatF-Drop and F-Match make the generators focus on fittingall of frequency components. More importantly, we foundthat F-Drop produces no visible flaws by filtering the high-frequency components during training. Additional visual-ization studies including ones on other datasets can be foundin the supplementary materials; they show the same ten-dency as described here.

7. ConclusionWe presented F-Drop and F-Match for minimizing the

frequency gaps that appear in images generated by GANs.We demonstrated that the discriminators of GANs arehighly sensitive to high-frequency perturbations and thesensitivity can cause frequency gaps. Our methods improveGANs in both the frequency and spatial domain because F-Drop protects the discriminators from high-frequency per-turbations and F-Match directly minimizes the frequencygap by using a simple mini-batch error function. Our exten-sive experiments show that the combination of F-Drop andF-Match outperforms the baselines on various datasets. Animportant direction of future research will be to introduceadaptive masking without any hyperparameter into F-Dropfor filtering effectively and generating realistic images.

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