Normalizing flows are an established approach for modelling complex probability densities through invertible transformations from a base distribution. However, the accuracy with which the target distribution can be captured by the normalizing flow is strongly influenced by the topology of the base distribution. A mismatch between the topology of the target and the base can result in a poor performance, as is typically the case for multi-modal problems. A number of different works have attempted to modify the topology of the base distribution to better match the target, either through the use of Gaussian Mixture Models (Izmailov et al., 2020; Ardizzone et al., 2020; Hagemann & Neumayer, 2021) or learned accept/reject sampling (Stimper et al., 2022). We introduce piecewise normalizing flows which divide the target distribution into clusters, with topologies that better match the standard normal base distribution, and train a series of flows to model complex multi-modal targets. We demonstrate the performance of the piecewise flows using some standard benchmarks and compare the accuracy of the flows to the approach taken in Stimper et al. (2022) for modelling multi-modal distributions. We find that our approach consistently outperforms the approach in Stimper et al. (2022) with a higher emulation accuracy on the standard benchmarks.

翻译：归一化流是一种通过从基分布进行可逆变换来建模复杂概率密度的成熟方法。然而，归一化流对目标分布的捕捉精度受到基分布拓扑结构的强烈影响。当目标与基分布拓扑结构不匹配时（例如多模态问题中的典型情况），可能导致性能较差。已有诸多研究尝试修改基分布拓扑以更好匹配目标：或采用高斯混合模型（Izmailov等，2020；Ardizzone等，2020；Hagemann & Neumayer，2021），或使用可学习的接受/拒绝采样（Stimper等，2022）。我们提出分段归一化流，该方法将目标分布划分为若干簇，使各簇的拓扑结构更匹配标准正态基分布，并训练一系列流模型来建模复杂多模态目标。我们通过若干标准基准测试展示了分段流的性能，并将其在建模多模态分布时的精度与Stimper等（2022）的方法进行对比。结果表明，我们的方法在标准基准测试中始终优于Stimper等（2022）的方法，具有更高的仿真精度。