We propose conformal generative modeling, a framework for generative modeling on 2D surfaces approximated by discrete triangle meshes. Our approach leverages advances in discrete conformal geometry to develop a map from a source triangle mesh to a target triangle mesh of a simple manifold such as a sphere. After accounting for errors due to the mesh discretization, we can use any generative modeling approach developed for simple manifolds as a plug-and-play subroutine. We demonstrate our framework on multiple complicated manifolds and multiple generative modeling subroutines, where we show that our approach can learn good estimates of distributions on meshes from samples, and can also learn simultaneously from multiple distinct meshes of the same underlying manifold.

翻译：我们提出共形生成建模——一种在由离散三角形网格近似的二维表面上进行生成建模的框架。该方法利用离散共形几何的进展，构建从源三角形网格到简单流形（如球面）的目标三角形网格的映射。在修正网格离散化造成的误差后，可将任何针对简单流形的生成建模方法作为即插即用于程序使用。我们在多个复杂流形及多种生成建模子程序上验证该框架，结果表明该方法能够从样本中学习网格上分布的良好估计，并能同时从同一底层流形的多个不同网格中进行学习。