In this paper, we present a novel localized Generative Adversarial Net (GAN) to learn on the manifold of real data. Compared with the classic GAN that {\em globally} parameterizes a manifold, the Localized GAN (LGAN) uses local coordinate charts to parameterize distinct local geometry of how data points can transform at different locations on the manifold. Specifically, around each point there exists a {\em local} generator that can produce data following diverse patterns of transformations on the manifold. The locality nature of LGAN enables local generators to adapt to and directly access the local geometry without need to invert the generator in a global GAN. Furthermore, it can prevent the manifold from being locally collapsed to a dimensionally deficient tangent subspace by imposing an orthonormality prior between tangents. This provides a geometric approach to alleviating mode collapse at least locally on the manifold by imposing independence between data transformations in different tangent directions. We will also demonstrate the LGAN can be applied to train a robust classifier that prefers locally consistent classification decisions on the manifold, and the resultant regularizer is closely related with the Laplace-Beltrami operator. Our experiments show that the proposed LGANs can not only produce diverse image transformations, but also deliver superior classification performances.

本文提出了一个新颖的局部生成对抗网络（Localized GAN），使用局部坐标系向量化真实数据的不同局部几何变换。在正交先验的作用下避免了流形局部坍塌到低维切向子空间，降低了模式崩溃的风险。用提出的LGAN训练分类器不仅能获得更优结果，而且分类结果是在流形中的局部连续性解释，与拉普拉斯-贝尔特拉米算子密切相关。

全局与本地化生成对抗网络