Signature and Log-Signature for the Study of Empirical Distributions Generated with GANs

Abstract

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In this paper, we address the research gap in efficiently assessing Generative Adversarial Network (GAN) convergence and goodness of fit by introducing the application of the Signature Transform to measure similarity between image distributions. Specifically, we propose the novel use of Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) Signature, along with Log-Signature, as alternatives to existing methods such as Fréchet Inception Distance (FID) and Multi-Scale Structural Similarity Index Measure (MS-SSIM). Our approach offers advantages in terms of efficiency and effectiveness, providing a comprehensive understanding and extensive evaluations of GAN convergence and goodness of fit. Furthermore, we present innovative analytical measures based on statistics by means of Kruskal–Wallis to evaluate the goodness of fit of GAN sample distributions. Unlike existing GAN measures, which are based on deep neural networks and require extensive GPU computations, our approach significantly reduces computation time and is performed on the CPU while maintaining the same level of accuracy. Our results demonstrate the effectiveness of the proposed method in capturing the intrinsic structure of the generated samples, providing meaningful insights into GAN performance. Lastly, we evaluate our approach qualitatively using Principal Component Analysis (PCA) and adaptive t-Distributed Stochastic Neighbor Embedding (t-SNE) for data visualization, illustrating the plausibility of our method.