Generative Adversarial Networks (GANs) have become a powerful framework to learn generative models that arise across a wide variety of domains. While there has been a recent surge in the development of numerous GAN architectures with distinct optimization metrics, we are still lacking in our understanding on how far away such GANs are from optimality. In this paper, we make progress on a theoretical understanding of the GANs under a simple linear-generator Gaussian-data setting where the optimal maximum-likelihood generator is known to perform Principal Component Analysis (PCA). We find that the original GAN by Goodfellow et. al. fails to recover the optimal PCA solution. On the other hand, we show that Wasserstein GAN can approach the PCA solution in the limit of sample size, and hence it may serve as a basis for an optimal GAN architecture that yields the optimal generator for a wide range of data settings.
Log-PCA versus Geodesic PCA of histograms in the Wasserstein space
本文研究了元素为随机直方图的数据集的统计分析。为了从这些数据中学习主要的变化模式，我们考虑了关于概率测度之间的2-瓦瑟斯坦距离计算直方图的PCA的问题。为此，我们建议比较Bigot et al.（2015）和Seguy and Cuturi（2015）介绍的Wasserstein空间中的log-PCA和测地线PCA方法。测地线PCA涉及解决非凸优化问题。为了近似求解，我们提出了一种新的前向-后向算法。这使得日志PCA和测地线PCA的一维直方图，我们进行使用各种数据集之间的详细比较，并强调每种方法的优点和缺点。我们将这些结果扩展到二维数据，并在该设置中比较这两种方法。
Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
Department of Statistics University of Illinois Urbana-Champaign（统计学系伊利诺伊大学厄巴纳-香槟分校）；Department of ACMS University of Notre Dame（ACMS系诺丁汉大学）；Department of Mathematics University of British Columbia（数学系不列颠哥伦比亚大学）；Department of Mathematics Thomas Lord Department of Computer Science University of Southern California（数学系托马斯·劳德计算机科学系南加州大学）
Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances
The Hong Kong Polytechnic University（香港理工大学）；Chinese Academy of Sciences（中国科学院）；Academy of Mathematics and Systems Science（数学系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
On the Wasserstein Geodesic Principal Component Analysis of probability measures
CREST-ENSAE, IP Paris（CREST-ENSAE，IP巴黎）；CNRS, IRIT, Université de Toulouse（CNRS，IRIT，图卢兹大学）；Université Paris 1 Panthéon Sorbonne（巴黎第一大学巴黎政治学院）；ENAC, IMT, Université de Toulouse（ENAC，IMT，图卢兹大学）
On optimal solutions of classical and sliced Wasserstein GANs with non-Gaussian data
Dept. of Applied Math., Univ. of Colorado, Boulder, CO 80309, USA（应用数学系，科罗拉多大学，伯尔德，科罗拉多州，80309，美国）；Department of EE and GICE, National Taiwan University, Taipei, Taiwan（电子工程系和信息科学与工程系，国立台湾大学，台北，台湾）；Institute of Communications Engineering, National Yang Ming Chiao Tung University, HsinChu, Taiwan（通讯工程研究所，国立阳明交通大学，新竹，台湾）；Bosch Center for AI, USA（博世人工智能中心，美国）
