Normalizing flows (NF) use a continuous generator to map a simple latent (e.g. Gaussian) distribution, towards an empirical target distribution associated with a training data set. Once trained by minimizing a variational objective, the learnt map provides an approximate generative model of the target distribution. Since standard NF implement differentiable maps, they may suffer from pathological behaviors when targeting complex distributions. For instance, such problems may appear for distributions on multi-component topologies or characterized by multiple modes with high probability regions separated by very unlikely areas. A typical symptom is the explosion of the Jacobian norm of the transformation in very low probability areas. This paper proposes to overcome this issue thanks to a new Markov chain Monte Carlo algorithm to sample from the target distribution in the latent domain before transporting it back to the target domain. The approach relies on a Metropolis adjusted Langevin algorithm (MALA) whose dynamics explicitly exploits the Jacobian of the transformation. Contrary to alternative approaches, the proposed strategy preserves the tractability of the likelihood and it does not require a specific training. Notably, it can be straightforwardly used with any pre-trained NF network, regardless of the architecture. Experiments conducted on synthetic and high-dimensional real data sets illustrate the efficiency of the method.

翻译：归一化流（NF）利用连续生成器将简单潜变量（如高斯分布）映射到与训练数据集相关的经验目标分布。通过最小化变分目标进行训练后，学习到的映射为目标分布提供了近似生成模型。由于标准NF实现可微映射，当应对复杂分布时可能出现病态行为。例如，在多分量拓扑结构或由极低概率区域分隔的高概率多模态分布中可能显现此类问题。典型症状是极低概率区域变换的雅可比范数爆炸。本文提出通过新型马尔可夫链蒙特卡洛算法解决此问题：在潜域对目标分布进行采样后，再将其映射回目标域。该方法采用显式利用变换雅可比矩阵动力学的梅特罗波利斯调整朗之万算法（MALA）。与其他方法相比，所提策略保持似然可计算性且无需特定训练。值得注意的是，无论采用何种架构，该方法均可直接用于任何预训练NF网络。基于合成数据和高维真实数据集的实验验证了该方法的有效性。