Injective flows for star-like manifolds
This work addresses a computational bottleneck in density estimation on manifolds for researchers in machine learning and statistics, though it is incremental as it builds on existing injective flow methods.
The authors tackled the computational challenge of computing the Jacobian determinant for injective flows on manifolds by proposing a method for star-like manifolds that allows exact and efficient computation, matching the cost of normalizing flows, and demonstrated its relevance in objective Bayesian approaches and probabilistic mixing models.
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 star-like manifolds and show that for such manifolds we can compute the Jacobian determinant exactly and efficiently, with the same cost as NFs. This aspect is particularly relevant for variational inference settings, where no samples are available and only some unnormalized target is known. Among many, we showcase the relevance of modeling densities on star-like manifolds in two settings. Firstly, we introduce a novel Objective Bayesian approach for penalized likelihood models by interpreting level-sets of the penalty as star-like manifolds. Secondly, we consider probabilistic mixing models and introduce a general method for variational inference by defining the posterior of mixture weights on the probability simplex.