在潜空间中使用 Langevin 动力学的 Normalizing flow 采样
Normalizing flow sampling with Langevin dynamics in the latent space
AI总结：
针对标准 Normalizing flows 在复杂多峰分布上因 Jacobian 范数爆炸导致的病态行为，提出一种基于 MALA 的 MCMC 算法，在潜空间采样后映射回目标域，无需特定训练即可兼容任何预训练 NF 网络。
摘要：
Normalizing flows (NF) 使用连续生成器将简单的潜分布（例如 Gaussian 分布）映射到与训练数据集相关的经验目标分布。一旦通过最小化变分目标完成训练，学习到的映射便提供了目标分布的近似生成模型。由于标准 NF 实现的是可微映射，当目标分布复杂时，它们可能会出现病态行为。例如，对于具有多组件拓扑结构，或以被极低概率区域分隔的多个高概率区域为特征的多峰分布，此类问题就会出现。一个典型症状是变换的 Jacobian 范数在极低概率区域发生爆炸。本文提出通过一种新的 Markov chain Monte Carlo 算法来解决此问题，该算法先在潜空间中对目标分布进行采样，随后再将其传输回目标域。该方法依赖于 Metropolis adjusted Langevin algorithm (MALA)，其动力学明确利用了变换的 Jacobian。与替代方法相反，所提出的策略保留了似然的可处理性，且不需要特定训练。值得注意的是，它可以直接用于任何预训练的 NF 网络，而无需考虑其架构。在合成和高维真实数据集上进行的实验证明了该方法的有效性。
英文摘要：
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.
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