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1. SDYN-GANs: Adversarial learning methods for multistep generative models for general order stochastic dynamics

   https://doi.org/10.1016/j.jcp.2024.113442  

   Stinis, P; Daskalakis, C; Atzberger, PJ (December 2024, Journal of Computational Physics)

   We introduce adversarial learning methods for data-driven generative modeling of dynamics of n-th-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable m-step stochastic numerical integrators. From observations of trajectory samples, we introduce methods for learning long-time predictors and stable representations of the dynamics. Our approaches use discriminators based on Maximum Mean Discrepancy (MMD), training protocols using both conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. Our adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems. 

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2. Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit

   Zhu, Shengyu; Chen, Biao; Yang, Pengfei; Chen, Zhitang (April 2019, International Conference on Artificial Intelligence and Statistics)

   We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on Rd, while the quadratictime Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov’s theorem from large deviation theory and the weak metrizable properties of the MMD and KSD. 

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3. Joint Generative Moment-Matching Network for Learning Structural Latent Code

   https://doi.org/10.24963/ijcai.2018/293  

   Gao, Hongchang; Huang, Heng (July 2018, 27th International Joint Conference on Artificial Intelligence (IJCAI 2018))

   Generative Moment-Matching Network (GMMN) is a deep generative model, which employs maximum mean discrepancy as the objective to learn model parameters. However, this model can only generate samples, failing to infer the latent code from samples for downstream tasks. In this paper, we propose a novel Joint Generative Moment-Matching Network (JGMMN), which learns the structural latent code for unsupervised inference. Specifically, JGMMN has a generation network for the generation task and an inference network for the inference task. We first reformulate this model as the two joint distributions matching problem. To solve this problem, we propose to use the Joint Maximum Mean Discrepancy (JMMD) as the objective to learn these two networks simultaneously. Furthermore, to enforce the consistency between the sample distribution and the inferred latent code distribution, we propose a novel multi-modal regularization to enforce this consistency. At last, extensive experiments on both synthetic and real-world datasets have verified the effectiveness and correctness of our proposed JGMMN. 

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4. An empirical study on evaluation metrics of generative adversarial networks

   Xu, Q; Huang, G; Yan, Y; Guo, C; Sun, Y; Wu, F; Weinberger, K (April 2018, arXiv.org)

   Evaluating generative adversarial networks (GANs) is inherently challenging. In this paper, we revisit several representative sample-based evaluation metrics for GANs, and address the problem of how to evaluate the evaluation metrics. We start with a few necessary conditions for metrics to produce meaningful scores, such as distinguishing real from generated samples, identifying mode dropping and mode collapsing, and detecting overfitting. With a series of carefully designed experiments, we comprehensively investigate existing sample-based metrics and identify their strengths and limitations in practical settings. Based on these results, we observe that kernel Maximum Mean Discrepancy (MMD) and the 1-Nearest- Neighbor (1-NN) two-sample test seem to satisfy most of the desirable properties, provided that the distances between samples are computed in a suitable feature space. Our experiments also unveil interesting properties about the behavior of several popular GAN models, such as whether they are memorizing training samples, and how far they are from learning the target distribution. 

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5. Characteristic and Universal Tensor Product Kernels

   Szabo, Z; Sriperumbudur, B. (January 2018, Journal of machine learning research)

   Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of the characteristic property of the tensor product kernel. 

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