论文标题
神经普通微分方程
Neural Ordinary Differential Equations on Manifolds
论文作者
论文摘要
归一化流是从复杂的多模式分布中获得可重新聚集样品的强大技术。不幸的是，当基础空间具有非琐碎的拓扑结构，并且仅适用于最基本的几何形状时，目前的方法缺乏。最近，基于神经odes的欧几里得空间中的流动标准化表现出了很大的希望，但遭受了相同的局限性。利用差异几何和几何控制理论中的思想，我们描述了如何将神经源扩展到光滑的歧管。我们展示了向量字段如何提供一个通用框架，用于在这些空间上参数化灵活的可逆映射类别，并说明如何进行基于梯度的学习。结果，我们定义了一种在歧管上构建归一流流量的一般方法。
Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only available for the most basic geometries. Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. Using ideas from differential geometry and geometric control theory, we describe how neural ODEs can be extended to smooth manifolds. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces and we illustrate how gradient based learning can be performed. As a result we define a general methodology for building normalizing flows on manifolds.
