# Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data

arXiv: https://arxiv.org/abs/2601.18728 · HTML: https://mlsanity.com/paper/2601.18728/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 60 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 30 of 36)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [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)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Dynamic CT Reconstruction from Limited Views with Implicit Neural Representations and Parametric Motion Fields](https://mlsanity.com/paper/2104.11745/) (2021)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data](https://mlsanity.com/paper/2404.10177/) (2024)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [An Expectation-Maximization Algorithm for Training Clean Diffusion Models from Corrupted Observations](https://mlsanity.com/paper/2407.01014/) (2024)
- [Fast and Provable ADMM for Learning with Generative Priors](https://mlsanity.com/paper/1907.03343/) (2019)
- [How much is a noisy image worth? Data Scaling Laws for Ambient Diffusion](https://mlsanity.com/paper/2411.02780/) (2024)
- [Ambient Diffusion Omni: Training Good Models with Bad Data](https://mlsanity.com/paper/2506.10038/) (2025)
- [Task-Aware Compressed Sensing with Generative Adversarial Networks](https://mlsanity.com/paper/1802.01284/) (2018)
- [Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusion](https://mlsanity.com/paper/2502.21278/) (2025)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [AmbientFlow: Invertible generative models from incomplete, noisy measurements](https://mlsanity.com/paper/2309.04856/) (2023)
- [Efficient Prior Calibration From Indirect Data](https://mlsanity.com/paper/2405.17955/) (2024)
- [Stochastic Forward-Backward Deconvolution: Training Diffusion Models with Finite Noisy Datasets](https://mlsanity.com/paper/2502.05446/) (2025)
- [Learning Distances from Data with Normalizing Flows and Score Matching](https://mlsanity.com/paper/2407.09297/) (2024)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Pullback Flow Matching on Data Manifolds](https://mlsanity.com/paper/2410.04543/) (2024)
- [Denoising Score Distillation: From Noisy Diffusion Pretraining to One-Step High-Quality Generation](https://mlsanity.com/paper/2503.07578/) (2025)
- [Curvature corrected tangent space-based approximation of manifold-valued data](https://mlsanity.com/paper/2306.00507/) (2023)

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

- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)

## Abstract (arXiv)

Modern generative modeling methods have demonstrated strong performance in learning complex data distributions from clean samples. In many scientific and imaging applications, however, clean samples are unavailable, and only noisy or linearly corrupted measurements can be observed. Moreover, latent structures, such as manifold geometries, present in the data are important to extract for further downstream scientific analysis. In this work, we introduce Riemannian AmbientFlow, a framework for simultaneously learning a probabilistic generative model and the underlying, nonlinear data manifold directly from corrupted observations. Building on the variational inference framework of AmbientFlow, our approach incorporates data-driven Riemannian geometry induced by normalizing flows, enabling the extraction of manifold structure through pullback metrics and Riemannian Autoencoders. We establish theoretical guarantees showing that, under appropriate geometric regularization and measurement conditions, the learned model recovers the underlying data distribution up to a controllable error and yields a smooth, bi-Lipschitz manifold parametrization. We further show that the resulting smooth decoder can serve as a principled generative prior for inverse problems with recovery guarantees. We empirically validate our approach on low-dimensional synthetic manifolds and on MNIST.