arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2011.12087cs.LGmath.STstat.TH
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一个方便的无限维生成对抗学习框架

A Convenient Infinite Dimensional Framework for Generative Adversarial Learning

University of Wuppertal(伍珀塔尔大学);Institute of Mathematical Modelling, Analysis and Computational Mathematics(数学建模、分析与计算数学研究所);Interdisciplinary Center for Machine Learning and Data Analytics(机器学习与数据分析跨学科中心)

Hayk Asatryan, Hanno Gottschalk, Marieke Lippert, Matthias Rottmann

摘要:

近年来,生成对抗网络(GANs)已经展示了令人印象深刻的实验结果,而只有少数作品为GANs培养统计学习理论。在这项工作中,我们提出了一个生成对抗学习的无限维理论框架。我们假设基本测度的概率密度函数是一致有界的,$k$ -倍$α$ -Hölder可微的($C^{k,α}$),并且一致有界远离零。在这些假设下,我们证明了Rosenblatt变换诱导出一个最优生成元,该最优生成元在$C^{k,α}$ -生成元的假设空间中是可实现的.通过对判别器的假设空间的一致定义,我们进一步证明了由对抗学习过程中的生成器诱导的分布与数据生成分布之间的Jensen-Shannon散度收敛到零。在数据生成过程的密度上的某些规律性假设下,我们还提供了基于链接和集中的收敛速率。

英文摘要:

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, $k$-times $α$-Hölder differentiable ($C^{k,α}$) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of $C^{k,α}$-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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