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"content": "Abdul Fatir Ansari†, Jonathan Scarlett†‡, and Harold Soh† \n†Department of Computer Science \n‡Department of Mathematics \nNational University of Singapore \n{afatir, scarlett, harold}@comp.nus.edu.sg"
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"content": "Implicit Generative Models (IGMs) such as GANs have emerged as effective data-driven models for generating samples, particularly images. In this paper, we formulate the problem of learning an IGM as minimizing the expected distance between characteristic functions. Specifically, we minimize the distance between characteristic functions of the real and generated data distributions under a suitably-chosen weighting distribution. This distance metric, which we term as the characteristic function distance (CFD), can be (approximately) computed with linear time-complexity in the number of samples, in contrast with the quadratic-time Maximum Mean Discrepancy (MMD). By replacing the discrepancy measure in the critic of a GAN with the CFD, we obtain a model that is simple to implement and stable to train. The proposed metric enjoys desirable theoretical properties including continuity and differentiability with respect to generator parameters, and continuity in the weak topology. We further propose a variation of the CFD in which the weighting distribution parameters are also optimized during training; this obviates the need for manual tuning, and leads to an improvement in test power relative to CFD. We demonstrate experimentally that our proposed method outperforms WGAN and MMD-GAN variants on a variety of unsupervised image generation benchmarks."
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"content": "Implicit Generative Models (IGMs), such as Generative Adversarial Networks (GANs) [12], seek to learn a model "
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"content": ". Unlike prescribed probabilistic models, IGMs do not require a likelihood function, and thus are appealing when the data likelihood is unknown or intractable. Empirically, GANs have excelled at numerous tasks, from unsupervised image generation [18] to policy learning [17]."
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"content": "tricks to stabilize training [31]. Subsequent research has revealed that the generator-discriminator setup in the GAN minimizes the Jensen-Shannon divergence between the real and generated data distributions; this divergence possesses discontinuities that results in uninformative gradients as "
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"content": ", which hampers training. Various works have since established desirable properties for a divergence that can ease GAN training, and proposed alternative training schemes [2, 34, 3], primarily using distances belonging to the Integral Probability Metric (IPM) family [29]. One popular IPM is the kernel-based metric Maximum Mean Discrepancy (MMD), and a significant portion of recent work has focussed on deriving better MMD-GAN variants [21, 5, 1, 22]."
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"content": "In this paper, we undertake a different, more elementary approach, and formulate the problem of learning an IGM as minimizing the expected distance between characteristic functions of real and generated data distributions. Characteristic functions are widespread in probability theory and have been used for two-sample testing [15, 11, 8], yet surprisingly, have not yet been investigated for GAN training. We find that this approach leads to a simple and computationally-efficient loss: the characteristic function distance (CFD). Computing CFD requires linear time in the number of samples (unlike the quadratic-time MMD), and our experimental results indicate that CFD minimization results in effective training."
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"content": "This work provides both theoretical and empirical support for using CFD to train IGMs. We first establish that the CFD is continuous and differentiable almost everywhere with respect to the parameters of the generator, and that it satisfies continuity in the weak topology - key properties that make it a suitable GAN metric [3, 21]. We provide novel direct proofs that supplement the existing theory on GAN training metrics. Algorithmically, our key idea is simple: train GANs using empirical estimates of the CFD under optimized weighting distributions. We report on systematic experiments using synthetic distributions and four benchmark image datasets (MNIST, CIFAR10, STL10, CelebA)."
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"content": " that have the same mean in all dimensions except one. As the dimensionality increases, it becomes increasingly difficult to distinguish between samples from the two distributions. In our tests, the weighting distribution "
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"content": " was set to 3. We optimized the parameter vector "
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"content": " to maximize the ECFD using the Adam optimizer for 100 iterations with a batch-size of 1000."
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"content": "Fig. 1a shows the variation of the test power (i.e., the fraction of times the null hypothesis "
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"content": " is rejected) with the number of dimensions. OEFCD refers to the optimized ECFD, and the \"Smooth\" suffix indicates the smoothed ECFD variant proposed by [8]. We see that optimization of "
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"content": " increases the power of ECFD and ECFD-Smooth, particularly at the higher dimensionalities. There do not appear to be significant differences between the optimized smoothed and non-smoothed ECFD variants. Moreover, the optimization improved the ability of the test to correctly distinguish the two different distributions, but did not hamper its ability to correctly accept the null hypothesis when the distributions are the same (see Appendix C)."
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"content": "Similar to recently proposed Wasserstein [3] and MMD [21] GANs, the CFD exhibits desirable mathematical properties. Specifically, CFD is continuous and differentiable almost everywhere in the parameters of the generator (Thm. 1). Moreover, as it is continuous in the weak topology (Thm. 2), it can provide a signal to the generator "
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"content": "lack this property (e.g., Jensen-Shannon divergence). In the following, we provide proofs for the above claims under assumptions similar to [3]."
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"content": ". In this result, we let "
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"content": " implies convergence in distribution."
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"content": " under the distributions considered. The Lipschitz properties ensure that the function difference vanishes when one distribution converges to the other."
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"content": " is a feed-forward network with ReLU activations. To ensure that "
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"content": " is Lipschitz, common methods employed in prior work include weight clipping [3] and gradient penalty [14]. In addition, many common distributions satisfy "
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"content": "\\mathbb{E}_{\\omega(\\mathbf{t})}[\\| \\mathbf{t} \\|] < \\infty"
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"content": "An example demonstrating the necessity of Lipschitz assumptions in continuity results (albeit for a different metric) can be found in Example 1 of [1]. In the appendix, we discuss conditions under which Theorem 2 can be strengthened to an \"if and only if\" statement."
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"content": "Learning a generative model by minimizing the MMD between real and generated samples was proposed independently by [23] and [10]. The Generative Moment Matching Network (GMMN) [23] uses an autoencoder to first transform the data into a latent space, and then trains a generative network to produce latent vectors that match the true latent distribution. The MMD-GAN [21] performs a similar input transformation using a network "
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"content": " and the generator distribution "
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"content": "; this results in a GAN-like min-max criterion. More recently, [5] and [1] have proposed different theoretically-motivated regularizers on the gradient of MMD-GAN critic that improve training. In our experiments, we compare against the MMD-GAN both with and without gradient regularization."
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"content": "Very recent work [22] (IKL-GAN) has evaluated kernels parameterized in Fourier space, which are then used to compute MMD in MMD-GAN. In contrast to IKL-GAN, we derive the CF-GAN via characteristic functions rather than via MMD, and our method obviates the need for kernel evaluation. We also provide novel direct proofs for the theoretical properties of the optimized CFD that are not based on its equivalence to MMD. The IKL-GAN utilizes a neural network to sample random frequencies, whereas we use a simpler fixed distribution with a learned scale, reducing the number of hyperparameters to tune. Our method yields state-of-the-art performance, which suggests that the more complex setup in IKL-GAN may not be required for effective GAN training."
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"content": "In parallel, significant work has gone into improving GAN training via architectural and optimization enhancements [27, 7, 18]; these research directions are orthogonal to our work and can be incorporated in our proposed model."
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"content": " was kept fixed. Similarly, we suffix GP to the model name when gradient penalty [14] was used to enforce Lipschitzness of "
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"content": "f_{\\phi}"
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"content": ". In the absence of gradient penalty, we clipped the weights of "
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"content": " in "
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"content": "[-0.01, 0.01]"
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"type": "text",
"content": ". When the parameters "
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"content": "\\sigma"
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"content": " were optimized, we scaled the ECFD by "
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"content": "\\|\\sigma\\|"
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"content": " to prevent "
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"content": " from going to infinity, thereby ensuring "
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"content": "\\mathbb{E}_{\\omega(\\mathbf{t})}[\\|\\mathbf{t}\\|] < \\infty"
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"type": "text",
"content": "."
}
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"content": "We compare our proposed model against two variants of MMD-GAN: (i) MMD-GAN [21], which uses MMD with a mixture of RBF kernels as the distance metric; (ii) MMD-GAN- "
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"type": "text",
"content": " [5], which introduces an additive gradient penalty based on MMD's IPM witness function, an L2 penalty on discriminator activations, and uses a mixture of RQ kernels. We also compare against WGAN [3] and WGAN-GP [14] due to their close relation to MMD-GAN [21, 5]. Our code is available online at https://github.com/crlslab/OCFGAN."
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"content": " and a more complex bimodal distribution "
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"content": ". The distributions were constructed by transforming "
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"content": "z \\sim \\mathcal{N}(0,1)"
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"content": " using a function "
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"content": "h: \\mathbb{R} \\to \\mathbb{R}"
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"content": ". For the unimodal dataset, we used the scale-shift function form used by [37], where "
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"content": "h(z) = \\mu + \\sigma z"
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"content": ". For the bimodal dataset, we used the function form used by planar flow [32], where "
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"content": "h(z) = \\alpha z + \\beta \\tanh(\\gamma \\alpha z)"
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"content": ". We trained the various GAN models to approximate the distribution of the transformed samples. Once trained, we compared the transformation function "
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"content": " learned by the GAN against the true function "
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"content": "h"
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"type": "text",
"content": ". We computed the mean absolute error (MAE) "
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"content": "(\\mathbb{E}_z[|h(z) - \\hat{h}(z)|])"
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"content": " to evaluate the models. Further details on the experimental setup are in Appendix B.1."
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"content": "Figs. 2a and 2b show the variation of the MAE with training iterations. For both datasets, the models with gradient penalty converge to better minima. In "
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"type": "text",
"content": ", MMD-GAN-GP and OCF-GAN-GP converge to the same value of MAE, but MMD-GAN-GP converges faster. During our experiments, we observed that the scale of the weighting distribution (which is initialized to 1) falls rapidly before the MAE begins to decrease. For the experiments with the scale fixed at 0.1 (CF-GAN-GP"
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"content": ") and 1 (CF-GAN-GP"
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"content": "), both models converge to the same MAE, but CF-GAN-GP"
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"content": " takes much longer to converge than CF-GAN-GP"
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"type": "text",
"content": ". This indicates that the optimization of the scale parameter leads to faster convergence. For the more complex dataset "
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"type": "text",
"content": ", MMD-GAN-GP takes significantly longer to converge compared to WGAN-GP and OCF-GAN-GP. OCF-GAN-GP converges fastest and to a better minimum,"
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"content": "A recent large-scale analysis of GANs [26] showed that different models achieve similar best performance when given ample computational budget, and advocates comparisons between distributions under practical settings. As such, we compare scores attained by the models from different initializations under fixed computational budgets. We used four datasets: 1) MNIST [20]: 60K grayscale images of handwritten digits; 2) CIFAR10 [19]: 50K RGB images; 3) CelebA [24]: "
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"content": "200K RGB images of celebrity faces; and 4) STL10 [9]: 100K RGB images. For all datasets, we center-cropped and scaled the images to "
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"content": "32 \\times 32"
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"content": "."
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"content": "Network and Hyperparameter Details Given our computational budget and experiment setup, we used a DCGAN-like generator "
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"type": "text",
"content": " architecture for all models (similar to [21]). For MMD-GAN, we used a mixture of five RBF kernels (5-RBF) with different scales [21]. MMD-GAN- "
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"content": "\\mathrm{GP}_{L2}"
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"type": "text",
"content": " used a mixture of rational quadratic kernels (5-RQ). The kernel parameters and the trade-off parameters for gradient and L2 penalties were set according to [5]. We tested CF-GAN variants with two weighting distributions: Gaussian "
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"content": "(\\mathcal{N})"
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"content": " and Student's-t "
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"content": "(\\mathcal{T})"
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"type": "text",
"content": " (with 2 degrees of freedom). For CF-GAN, we tested 3 scale parameters in the set "
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"content": "\\{0.2, 0.5, 1\\}"
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"content": ", and we report the best results. The number of frequencies "
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"content": "(k)"
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"content": " for computing ECFD was set to 8. Please see Appendix B.2 for implementation details."
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"content": "Evaluation Metrics We compare the different models using three evaluation metrics: Fréchet Inception Distance (FID) [34], Kernel Inception Distance (KID) [5], and Precision-Recall (PR) for generative models [33]. Details on these metrics and the evaluation procedure can be found in Appendix B.2. In brief, the FID computes the Fréchet distance between two multivariate Gaussians and the KID"
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"content": "computes the MMD (with a polynomial kernel of degree 3) between the real and generated data distributions. Both FID and KID give single value scores, and PR gives a two dimensional score which disentangles the quality of generated samples from the coverage of the data distribution. PR is defined by a pair "
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"content": "F_{8}"
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"type": "text",
"content": " (recall) and "
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"type": "inline_equation",
"content": "F_{1/8}"
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{
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"type": "text",
"content": " (precision) which represent the coverage and sample quality, respectively [33]."
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"content": "Results In the following, we summarize our main findings, and relegate the details to the Appendix. Table 1 shows the FID and KID values achieved by different models for CIFAR10, STL10, and CelebA datasets. In short, our model outperforms both variants of WGAN and MMDGAN by a significant margin. OCF-GAN, using just one weighting function, outperforms both MMD-GANs that use a mixture of 5 different kernels."
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"content": "We observe that the optimization of the scale parameter improves the performance of the models for both weighting distributions, and the introduction of gradient penalty as a means to ensure Lipschitzness of "
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"type": "text",
"content": " results in a significant improvement in the score values for all models. This is in line with the results of [14] and [5]. Overall, amongst the CF-GAN variants, OCF-GAN-GP with Gaussian weighting performs the best for all datasets."
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"content": "The two-dimensional precision-recall scores in Fig. 3 provide further insight into the performance of different models. Across all the datasets, the addition of gradient penalty (OCF-GAN-GP) rather than weight clipping (OCF-GAN) leads to a higher improvement in recall compared to precision. This result supports recent arguments that weight clipping forces the generator to learn simpler functions, while gradient penalty is more flexible [14]. The improvement in recall with the introduction of gradient penalty is more noticeable for CIFAR10 and STL10 datasets compared to CelebA. This result is intuitive; CelebA is a more uniform and simpler dataset compared to CIFAR10/STL10, which contain more diverse classes of images, and thus"
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"content": "likely have modes that are more complex and far apart. Results on the MNIST dataset, where all models achieve good score values, are available in Appendix C, which also includes further experiments using the smoothed version of ECFD and the optimized smoothed version (no improvement over the unsmoothed versions on the image datasets)."
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| WGAN | - | 44.11 (1.16) | 25 (1) | 38.61 (0.43) | 23 (1) | 17.85 (0.69) | 12 (1) |
| WGAN-GP | - | 35.91 (0.30) | 19 (1) | 27.85 (0.81) | 15 (1) | 10.03 (0.37) | 6 (1) |
| MMD-GAN | 5-RBF | 41.28 (0.54) | 23 (1) | 35.76 (0.54) | 21 (1) | 18.48 (1.60) | 12 (1) |
| MMD-GAN-GPL2 | 5-RQ | 38.88 (1.35) | 21 (1) | 31.67 (0.94) | 17 (1) | 13.22 (1.30) | 8 (1) |
| CF-GAN | N(σ=0.5) | 39.81 (0.93) | 23 (1) | 33.54 (1.11) | 19 (1) | 13.71 (0.50) | 9 (1) |
| T(σ=1) | 41.41 (0.64) | 22 (1) | 35.64 (0.44) | 20 (1) | 16.92 (1.29) | 11 (1) |
| OCF-GAN | N | 38.47 (1.00) | 20 (1) | 32.51 (0.87) | 19 (1) | 14.91 (0.83) | 9 (1) |
| T | 37.96 (0.74) | 20 (1) | 31.03 (0.82) | 17 (1) | 13.73 (0.56) | 8 (1) |
| OCF-GAN-GP | N | 33.08 (0.26) | 17 (1) | 26.16 (0.64) | 14 (1) | 9.39 (0.25) | 5 (1) |
| T | 34.33 (0.77) | 18 (1) | 26.86 (0.38) | 15 (1) | 9.61 (0.39) | 6 (1) |
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| 1 | 0.44 (0.03) | 5 (1) |
| 4 | 0.39 (0.05) | 4 (1) |
| 8 | 0.36 (0.03) | 4 (1) |
| 16 | 0.35 (0.02) | 3 (1) |
| 32 | 0.35 (0.03) | 3 (1) |
| 64 | 0.36 (0.07) | 4 (1) |
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"content": "In this paper, we proposed a novel weighted distance between characteristic functions for training IGMs, and showed that the proposed metric has attractive theoretical properties. We observed experimentally that the proposed model outperforms MMD-GAN and WGAN variants on four benchmark image datasets. Our results indicate that characteristic functions provide an effective alternative means for training IGMs."
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"content": "This work opens additional avenues for future research. For example, the empirical CFD used for training may result in high variance gradient estimates (particularly with a small number of sampled frequencies), yet the CFD-trained models attain high performance scores with better convergence in our tests. The reason for this should be more thoroughly explored. Although we used the gradient penalty proposed by WGAN-GP, there is no reason to constrain the gradient to exactly 1. We believe that an exploration of the geometry of the proposed loss could lead to improvement in the gradient regularizer for the proposed method."
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"content": "Apart from generative modeling, two sample tests such as MMD have been used for problems such as domain adaptation [25] and domain separation [6], among others. The optimized CFD loss function proposed in this work can be used as an alternative loss for these problems."
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"content": "Acknowledgements This research is supported by the National Research Foundation Singapore under its AI Singapore Programme (Award Number: AISG-RP-2019-011) to H. Soh. J. Scarlett is supported by the Singapore National Research Foundation (NRF) under grant number R-252-000-A74-281."
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