# Density estimation on low-dimensional manifolds: an inflation-deflation approach

arXiv: https://arxiv.org/abs/2105.12152 · HTML: https://mlsanity.com/paper/2105.12152/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 39 references that are in the map.
- First posted on arXiv 25 May 2021.

## Builds on (most cited first, 18 of 18)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [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)
- [SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds](https://mlsanity.com/paper/2006.04604/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Block Neural Autoregressive Flow](https://mlsanity.com/paper/1904.04676/) (2019)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Normalizing Flows Across Dimensions](https://mlsanity.com/paper/2006.13070/) (2020)

## Built on by (most cited first, 11 of 11)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Diagnosing and Fixing Manifold Overfitting in Deep Generative Models](https://mlsanity.com/paper/2204.07172/) (2022)
- [CaloMan: Fast generation of calorimeter showers with density estimation on learned manifolds](https://mlsanity.com/paper/2211.15380/) (2022)
- [Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity](https://mlsanity.com/paper/2505.09922/) (2025)
- [COMET Flows: Towards Generative Modeling of Multivariate Extremes and Tail Dependence](https://mlsanity.com/paper/2205.01224/) (2022)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Denoising Deep Generative Models](https://mlsanity.com/paper/2212.01265/) (2022)
- [Image Interpolation with Score-based Riemannian Metrics of Diffusion Models](https://mlsanity.com/paper/2504.20288/) (2025)
- [Manifold-Aware Perturbations for Constrained Generative Modeling](https://mlsanity.com/paper/2601.23151/) (2026)
- [Density Estimation via Binless Multidimensional Integration](https://mlsanity.com/paper/2407.08094/) (2024)
- [From Core to Detail: Unsupervised Disentanglement with Entropy-Ordered Flows](https://mlsanity.com/paper/2602.06940/) (2026)

## Abstract (arXiv)

Normalizing Flows (NFs) are universal density estimators based on Neural Networks. However, this universality is limited: the density's support needs to be diffeomorphic to a Euclidean space. In this paper, we propose a novel method to overcome this limitation without sacrificing universality. The proposed method inflates the data manifold by adding noise in the normal space, trains an NF on this inflated manifold, and, finally, deflates the learned density. Our main result provides sufficient conditions on the manifold and the specific choice of noise under which the corresponding estimator is exact. Our method has the same computational complexity as NFs and does not require computing an inverse flow. We also show that, if the embedding dimension is much larger than the manifold dimension, noise in the normal space can be well approximated by Gaussian noise. This allows using our method for approximating arbitrary densities on unknown manifolds provided that the manifold dimension is known.