We present a novel method for training score-based generative models which uses nonlinear noising dynamics to improve learning of structured distributions. Generalizing to a nonlinear drift allows for additional structure to be incorporated into the dynamics, thus making the training better adapted to the data, e.g., in the case of multimodality or (approximate) symmetries. Such structure can be obtained from the data by an inexpensive preprocessing step. The nonlinear dynamics introduces new challenges into training which we address in two ways: 1) we develop a new nonlinear denoising score matching (NDSM) method, 2) we introduce neural control variates in order to reduce the variance of the NDSM training objective. We demonstrate the effectiveness of this method on several examples: a) a collection of low-dimensional examples, motivated by clustering in latent space, b) high-dimensional images, addressing issues with mode collapse, small training sets, and approximate symmetries, the latter being a challenge for methods based on equivariant neural networks, which require exact symmetries.

翻译：我们提出了一种训练基于分数的生成模型的新方法，该方法利用非线性加噪动力学来改进对结构化分布的学习。将漂移项推广至非线性形式允许在动力学中融入额外的结构，从而使训练更好地适应数据特性，例如在多模态或（近似）对称性场景中。此类结构可通过低成本的预处理步骤从数据中获取。非线性动力学的引入为训练带来了新的挑战，我们通过两种方式应对：1）开发了一种新的非线性去噪分数匹配方法；2）引入神经控制变量以降低NDSM训练目标的方差。我们在多个示例中验证了该方法的有效性：a）受隐空间聚类启发的低维示例集合；b）针对高维图像数据，解决了模式坍塌、小训练集及近似对称性问题——后者对于基于等变神经网络的方法具有挑战性，因这类方法要求精确对称性。