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

TL;DR

Riemannian AmbientFlow introduces a novel framework that learns data manifolds and generative models directly from corrupted data, combining variational inference with Riemannian geometry to improve scientific data analysis and inverse problem solving.

Contribution

It develops a method that simultaneously learns data manifolds and generative models from noisy measurements using Riemannian geometry and variational inference, with theoretical recovery guarantees.

Findings

Successfully recovers underlying data distribution with controllable error.

Extracts smooth, bi-Lipschitz manifold parametrizations.

Demonstrates effectiveness on synthetic data and MNIST.

Abstract

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.