# Injective flows for star-like manifolds

arXiv: https://arxiv.org/abs/2406.09116 · HTML: https://mlsanity.com/paper/2406.09116/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 56 references that are in the map.
- First posted on arXiv 13 Jun 2024.

## Builds on (most cited first, 27 of 27)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [Regularized Autoencoders via Relaxed Injective Probability Flow](https://mlsanity.com/paper/2002.08927/) (2020)
- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [Invertible DenseNets with Concatenated LipSwish](https://mlsanity.com/paper/2102.02694/) (2021)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Probabilistic Archetypal Analysis](https://mlsanity.com/paper/1312.7604/) (2013)
- [Learning Extremal Representations with Deep Archetypal Analysis](https://mlsanity.com/paper/2002.00815/) (2020)
- [Jacobian Determinant of Normalizing Flows](https://mlsanity.com/paper/2102.06539/) (2021)
- [Lagrangian Flow Networks for Conservation Laws](https://mlsanity.com/paper/2305.16846/) (2023)
- [Conditional Matrix Flows for Gaussian Graphical Models](https://mlsanity.com/paper/2306.07255/) (2023)
- [Large-Scale Stochastic Sampling from the Probability Simplex](https://mlsanity.com/paper/1806.07137/) (2018)
- [Strategic Bayesian Asset Allocation](https://mlsanity.com/paper/1905.08414/) (2019)

## Built on by (most cited first, 1 of 1)

- [Marginal Flow: a flexible and efficient framework for density estimation](https://mlsanity.com/paper/2509.26221/) (2025)

## Abstract (arXiv)

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.