In this paper, we establish a connection between the parameterization of flow-based and energy-based generative models, and present a new flow-based modeling approach called energy-based normalizing flow (EBFlow). We demonstrate that by optimizing EBFlow with score-matching objectives, the computation of Jacobian determinants for linear transformations can be entirely bypassed. This feature enables the use of arbitrary linear layers in the construction of flow-based models without increasing the computational time complexity of each training iteration from $O(D^2L)$ to $O(D^3L)$ for an $L$-layered model that accepts $D$-dimensional inputs. This makes the training of EBFlow more efficient than the commonly-adopted maximum likelihood training method. In addition to the reduction in runtime, we enhance the training stability and empirical performance of EBFlow through a number of techniques developed based on our analysis of the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation while outperforming prior methods with a noticeable margin in terms of negative log-likelihood (NLL).

翻译：在本文中，我们建立了基于流和基于能量的生成模型参数化之间的联系，并提出了一种新的基于流的建模方法，称为能量归一化流（EBFlow）。我们证明，通过使用得分匹配目标优化EBFlow，可以完全避免线性变换的雅可比行列式计算。这一特性使得在构建基于流的模型时能够使用任意线性层，而不会将每个训练迭代的计算时间复杂度从$O(D^2L)$增加到$O(D^3L)$，其中$L$为层数，$D$为输入维度。这使得EBFlow的训练比常用的最大似然训练方法更加高效。除了减少运行时间外，我们还基于对得分匹配方法的分析，开发了一系列技术，提升了EBFlow的训练稳定性和经验性能。实验结果表明，我们的方法相比最大似然估计实现了显著的加速，同时在负对数似然（NLL）方面以明显优势超越了先前方法。