Based on the manifold hypothesis, real-world data often lie on a low-dimensional manifold, while normalizing flows as a likelihood-based generative model are incapable of finding this manifold due to their structural constraints. So, one interesting question arises: $\textit{"Can we find sub-manifold(s) of data in normalizing flows and estimate the density of the data on the sub-manifold(s)?"}$. In this paper, we introduce two approaches, namely per-pixel penalized log-likelihood and hierarchical training, to answer the mentioned question. We propose a single-step method for joint manifold learning and density estimation by disentangling the transformed space obtained by normalizing flows to manifold and off-manifold parts. This is done by a per-pixel penalized likelihood function for learning a sub-manifold of the data. Normalizing flows assume the transformed data is Gaussianizationed, but this imposed assumption is not necessarily true, especially in high dimensions. To tackle this problem, a hierarchical training approach is employed to improve the density estimation on the sub-manifold. The results validate the superiority of the proposed methods in simultaneous manifold learning and density estimation using normalizing flows in terms of generated image quality and likelihood.
翻译:基于多重假设, 真实世界的数据往往存在于一个低维的多元上, 而正常化的流量作为基于可能性的基因模型, 由于其结构性制约, 无法找到这一多重。 因此, 产生了一个有趣的问题 : $\ textit{ “ 我们能在正常流中找到数据分流的分流吗? $? 。 在本文中, 我们引入了两种方法, 即 per-pixel 受处罚的日志相似性和等级培训, 来回答上述问题 。 我们提出了一个单步方法, 用于通过拆分通过正常流到多重和场外部分获得的变换空间, 来联合多重学习和密度估计。 这样做的办法是, 由每平流来固定数据分流的可能性函数来计算。 正常流假定变换的数据是高的, 但这种强加的假设不一定是真实的, 特别是在高维度方面。 为了解决这个问题, 我们采用了分级培训方法来改进对亚安全度的密度估计。 使用正常质量估算的优越性方法, 并同时用多重学习的方式验证生成的密度的高度方法 。