黎曼环境流：面向从损坏数据同时进行流形学习和生成建模
Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data
AI总结：
提出Riemannian AmbientFlow框架，通过变分推断和数据驱动黎曼几何，从损坏观测中同时学习概率生成模型和非线性数据流形，并理论保证误差可控与双Lipschitz流形参数化。
摘要：
现代生成建模方法在从干净样本学习复杂数据分布方面表现出强大性能。然而，在许多科学和成像应用中，干净样本不可用，只能观测到噪声或线性损坏的测量值。此外，数据中存在的潜在结构（如流形几何）对于进一步的科学分析至关重要。在这项工作中，我们引入了Riemannian AmbientFlow，一个直接从损坏观测中同时学习概率生成模型和底层非线性数据流形的框架。基于AmbientFlow的变分推断框架，我们的方法结合了由归一化流引起的数据驱动黎曼几何，通过拉回度量和黎曼自编码器提取流形结构。我们建立了理论保证，表明在适当的几何正则化和测量条件下，学习到的模型以可控误差恢复底层数据分布，并产生光滑的双Lipschitz流形参数化。我们进一步证明，所得的光滑解码器可以作为具有恢复保证的逆问题的原则性生成先验。我们在低维合成流形和MNIST上实证验证了我们的方法。
英文摘要：
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.
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