Normalizing Flows (NFs) describe a class of models that express a complex target distribution as the composition of a series of bijective transformations over a simpler base distribution. By limiting the space of candidate transformations to diffeomorphisms, NFs enjoy efficient, exact sampling and density evaluation, enabling NFs to flexibly behave as both discriminative and generative models. Their restriction to diffeomorphisms, however, enforces that input, output and all intermediary spaces share the same dimension, limiting their ability to effectively represent target distributions with complex topologies. Additionally, in cases where the prior and target distributions are not homeomorphic, Normalizing Flows can leak mass outside of the support of the target. This survey covers a selection of recent works that combine aspects of other generative model classes, such as VAEs and score-based diffusion, and in doing so loosen the strict bijectivity constraints of NFs to achieve a balance of expressivity, training speed, sample efficiency and likelihood tractability.

翻译：标准化流 (Normalizing Flows, NFs) 描述了一类模型，这些模型将复杂的目标分布表示为一系列双射变换对简单基分布的复合结果。通过将候选变换空间限制为微分同胚，NFs 能够实现高效且精确的采样与密度评估，使其可灵活兼具判别模型与生成模型的双重功能。然而，对微分同胚的限制要求输入、输出及所有中间空间保持相同维度，这限制了其有效表示具有复杂拓扑结构的目标分布的能力。此外，当前验分布与目标分布非同胚时，标准化流可能导致质量泄露至目标支撑集之外。本综述选取了近期结合其他生成模型类别（如变分自编码器与基于分数的扩散模型）的研究工作，通过放宽NFs严格的双射性约束，在表达能力、训练速度、样本效率与似然可计算性之间实现平衡。