Normalizing Flows (NFs) are powerful and efficient models for density estimation. When modeling densities on manifolds, NFs can be generalized to injective flows but the Jacobian determinant becomes computationally prohibitive. Current approaches either consider bounds on the log-likelihood or rely on some approximations of the Jacobian determinant. In contrast, we propose injective flows for parametric hypersurfaces and show that for such manifolds we can compute the Jacobian determinant exactly and efficiently, with the same cost as NFs. Furthermore, we show that for the subclass of star-like manifolds we can extend the proposed framework to always allow for a Cartesian representation of the density. We showcase the relevance of modeling densities on hypersurfaces in two settings. Firstly, we introduce a novel Objective Bayesian approach to penalized likelihood models by interpreting level-sets of the penalty as star-like manifolds. Secondly, we consider Bayesian mixture models and introduce a general method for variational inference by defining the posterior of mixture weights on the probability simplex.

翻译：归一化流（NFs）是用于密度估计的强大且高效的模型。在对流形上的密度进行建模时，NFs可以推广为单射流，但此时雅可比行列式的计算变得难以处理。现有方法要么考虑对数似然的上界，要么依赖于对雅可比行列式的某些近似。相比之下，我们提出了针对参数化超曲面的单射流，并证明对于此类流形，我们可以精确且高效地计算雅可比行列式，其计算成本与NFs相同。此外，我们证明，对于星形流形这一子类，我们可以扩展所提出的框架，始终允许密度的笛卡尔坐标表示。我们在两种场景中展示了在超曲面上建模密度的相关性。首先，我们通过将惩罚项的等高线解释为星形流形，为惩罚似然模型引入了一种新颖的客观贝叶斯方法。其次，我们考虑贝叶斯混合模型，并通过在概率单纯形上定义混合权重的后验分布，引入了一种用于变分推断的通用方法。