# Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows

arXiv: https://arxiv.org/abs/2106.05275 · HTML: https://mlsanity.com/paper/2106.05275/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 75 references that are in the map.
- First posted on arXiv 9 Jun 2021.

## Builds on (most cited first, 30 of 49)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [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)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks](https://mlsanity.com/paper/1806.05393/) (2018)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [The Malicious Use of Artificial Intelligence: Forecasting, Prevention, and Mitigation](https://mlsanity.com/paper/1802.07228/) (2018)

## Built on by (most cited first, 30 of 31)

- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [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)
- [A Geometric Explanation of the Likelihood OOD Detection Paradox](https://mlsanity.com/paper/2403.18910/) (2024)
- [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)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [nu-Flows: Conditional Neutrino Regression](https://mlsanity.com/paper/2207.00664/) (2022)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Injectivity of ReLU networks: perspectives from statistical physics](https://mlsanity.com/paper/2302.14112/) (2023)
- [Denoising Deep Generative Models](https://mlsanity.com/paper/2212.01265/) (2022)
- [Flowification: Everything is a Normalizing Flow](https://mlsanity.com/paper/2205.15209/) (2022)
- [Neural Implicit Manifold Learning for Topology-Aware Density Estimation](https://mlsanity.com/paper/2206.11267/) (2022)
- [Spread Flows for Manifold Modelling](https://mlsanity.com/paper/2109.14216/) (2021)
- [Deterministic training of generative autoencoders using invertible layers](https://mlsanity.com/paper/2205.09546/) (2022)
- [Injectivity capacity of ReLU gates](https://mlsanity.com/paper/2410.20646/) (2024)
- [Deep Injective Prior for Inverse Scattering](https://mlsanity.com/paper/2301.03092/) (2023)
- [On Convolutions, Intrinsic Dimension, and Diffusion Models](https://mlsanity.com/paper/2506.20705/) (2025)
- [TR0N: Translator Networks for 0-Shot Plug-and-Play Conditional Generation](https://mlsanity.com/paper/2304.13742/) (2023)
- [Analyzing Generative Models by Manifold Entropic Metrics](https://mlsanity.com/paper/2410.19426/) (2024)
- [Conformal Generative Modeling on Triangulated Surfaces](https://mlsanity.com/paper/2303.10251/) (2023)
- [Injective flows for star-like manifolds](https://mlsanity.com/paper/2406.09116/) (2024)
- [Tunable Latent Generative Priors for Compressed Sensing and Inverse Problems](https://mlsanity.com/paper/2603.07357/) (2026)
- [Out-of-distribution detection using normalizing flows on the data manifold](https://mlsanity.com/paper/2308.13792/) (2023)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [Complex normalizing flows can almost be information K"ahler-Ricci flows](https://mlsanity.com/paper/2604.17954/) (2026)

## Abstract (arXiv)

Normalizing flows are generative models that provide tractable density estimation via an invertible transformation from a simple base distribution to a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and apply them in experiments with synthetic and real-world data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.