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 $\mathcal{O}(D^2L)$ to $\mathcal{O}(D^3L)$ 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 on the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation, while outperforming prior efficient training techniques with a noticeable margin in terms of negative log-likelihood (NLL).

翻译：本文建立了流式与能量生成模型参数化之间的联系，并提出了一种新型流建模方法——能量归一化流（EBFlow）。我们证明，通过使用分数匹配目标优化EBFlow，可以完全绕过线性变换中雅可比行列式的计算。这一特性使得在构建流式模型时能够使用任意线性层，且不会将接受D维输入的L层模型每次训练迭代的时间复杂度从$\mathcal{O}(D^2L)$增加至$\mathcal{O}(D^3L)$。这使得EBFlow的训练效率优于常用的最大似然训练方法。除运行时间缩短外，我们基于对分数匹配方法的分析开发了多项技术，显著提升了EBFlow的训练稳定性和实证性能。实验结果表明，本方法在负对数似然（NLL）指标上相较于最大似然估计实现了显著的加速效果，同时以明显优势超越了先前的高效训练技术。