arXiv:2601.18728v2 Announce Type: replace 
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.

تم تقديم إطار عمل جديد يسمى Riemannian AmbientFlow، والذي يهدف إلى تعلم نماذج توليد احتمالية والهندسة غير الخطية للبيانات مباشرة من الملاحظات الفاسدة. تعتمد هذه الطريقة على إطار الاستدلال التبايني لـ AmbientFlow وتدمج الهندسة الريمانية المدفوعة بالبيانات، مما يسهل استخراج هياكل المتنوعات من خلال تقنيات متقدمة مثل مقاييس السحب والمشفرات الذاتية الريمانية.

Se ha introducido un nuevo marco llamado Riemannian AmbientFlow, que tiene como objetivo aprender modelos generativos probabilísticos y la variedad de datos no lineales subyacente directamente a partir de observaciones corruptas. Este enfoque se basa en el marco de inferencia variacional de AmbientFlow e incorpora geometría riemanniana impulsada por datos, facilitando la extracción de estructuras de variedades a través de técnicas avanzadas como métricas de retroceso y autoencoders riemannianos.

Un nouveau cadre appelé Riemannian AmbientFlow a été introduit, visant à apprendre des modèles génératifs probabilistes et la structure sous-jacente des variétés non linéaires directement à partir d'observations corrompues. Cette approche s'appuie sur le cadre d'inférence variationnelle d'AmbientFlow et intègre la géométrie riemannienne induite par les flux normalisés, facilitant l'extraction des structures de variétés grâce à des techniques avancées telles que les métriques de rétroaction et les autoencodeurs riemanniens.

A new framework called Riemannian AmbientFlow has been introduced, which aims to learn probabilistic generative models and the underlying nonlinear data manifold directly from corrupted observations. This approach builds on the variational inference framework of AmbientFlow and incorporates data-driven Riemannian geometry, facilitating the extraction of manifold structures through advanced techniques like pullback metrics and Riemannian Autoencoders.