# Nonlinear Isometric Manifold Learning for Injective Normalizing Flows

arXiv: https://arxiv.org/abs/2203.03934 · HTML: https://mlsanity.com/paper/2203.03934/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 8 Mar 2022.

## Builds on (most cited first, 22 of 22)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [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)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [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)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Trumpets: Injective Flows for Inference and Inverse Problems](https://mlsanity.com/paper/2102.10461/) (2021)
- [LOCA: LOcal Conformal Autoencoder for standardized data coordinates](https://mlsanity.com/paper/2004.07234/) (2020)
- [Normalizing Flows Across Dimensions](https://mlsanity.com/paper/2006.13070/) (2020)
- [Isometric Autoencoders](https://mlsanity.com/paper/2006.09289/) (2020)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Density estimation on low-dimensional manifolds: an inflation-deflation approach](https://mlsanity.com/paper/2105.12152/) (2021)
- [Principal Component Density Estimation for Scenario Generation Using Normalizing Flows](https://mlsanity.com/paper/2104.10410/) (2021)
- [Principal Manifold Flows](https://mlsanity.com/paper/2202.07037/) (2022)

## Built on by (most cited first, 6 of 6)

- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Deterministic training of generative autoencoders using invertible layers](https://mlsanity.com/paper/2205.09546/) (2022)
- [Analyzing Generative Models by Manifold Entropic Metrics](https://mlsanity.com/paper/2410.19426/) (2024)
- [FloDR: An invertible dimensionality reduction method based on a normalising flow](https://mlsanity.com/paper/2607.26278/) (2026)
- [NAE: Normalizing AutoEncoder](https://mlsanity.com/paper/2608.12084/) (2026)

## Abstract (arXiv)

To model manifold data using normalizing flows, we employ isometric autoencoders to design embeddings with explicit inverses that do not distort the probability distribution. Using isometries separates manifold learning and density estimation and enables training of both parts to high accuracy. Thus, model selection and tuning are simplified compared to existing injective normalizing flows. Applied to data sets on (approximately) flat manifolds, the combined approach generates high-quality data.