AmbientFlow: Invertible Generative Modeling

• AmbientFlow is a variational inference framework that learns invertible generative models from noisy and incomplete measurements, enabling robust imaging and inverse problem solutions.
• It utilizes normalizing flows combined with importance-weighted Bayesian bounds to match observed measurement distributions without relying on clean training data.
• The Riemannian extension extracts explicit nonlinear manifold geometry, supporting enhanced probabilistic inversion and improved reconstruction quality.

AmbientFlow is a family of variational inference frameworks for learning invertible generative models directly from incomplete and noisy measurement data, with applications in imaging science, inverse problems, and manifold learning. The suite includes core AmbientFlow, which enables flow-based priors from corrupt measurements, and its Riemannian extension, which simultaneously learns an explicit nonlinear data manifold equipped with a data-driven geometry. The methodology exploits tractable likelihood evaluation in normalizing flows and variational Bayesian bounds to enable unsupervised prior learning and downstream probabilistic inversion, even when clean training data are absent (Kelkar et al., 2023, Diepeveen et al., 26 Jan 2026).

1. Problem Formulation

AmbientFlow addresses the unsupervised learning of an invertible generative model for an unknown object (or image) distribution $q_f(f)$, given only access to noisy, linear measurements

$g = H f + \eta, \quad \eta \sim q_n(\eta),$

where $H \in \mathbb{R}^{m \times n}$ is a known (possibly ill-posed) linear operator, and $\eta$ is additive noise with known density. The goal is to learn a flow-based generative model $p_\theta(f)$ that approximates $q_f(f)$ using only measurement samples $\{g^{(i)}\}_{i=1}^D$, exploiting the induced measurement distribution

$q_g(g) = \int q_n(g - Hf)\, q_f(f) \, df,$

making the inversion from measurements alone nontrivial (Kelkar et al., 2023).

The Riemannian AmbientFlow generalization assumes clean data concentrate near a low-dimensional, smooth manifold $\mathcal{M} \subset \mathbb{R}^D$. The observed measurements

$$y = A x + n, \quad x \sim p_\text{data}(x), \quad n \sim p_\text{noise},$$

reflect corrupted versions of latent clean objects. The challenge is to jointly estimate the data distribution and an explicit nonlinear manifold parameterization $g = H f + \eta, \quad \eta \sim q_n(\eta),$0 from corrupted observations (Diepeveen et al., 26 Jan 2026).

2. Variational Bayesian Objective and Invertible Flow Architecture

The central innovation of AmbientFlow is the shift from modeling unobserved $g = H f + \eta, \quad \eta \sim q_n(\eta),$1 directly to matching the induced measurement marginal. Let $g = H f + \eta, \quad \eta \sim q_n(\eta),$2 denote the invertible flow prior and $g = H f + \eta, \quad \eta \sim q_n(\eta),$3 an auxiliary invertible posterior flow parameterized by $g = H f + \eta, \quad \eta \sim q_n(\eta),$4. The modeled measurement marginal is

$g = H f + \eta, \quad \eta \sim q_n(\eta),$5

The training objective minimizes the Kullback–Leibler divergence between empirical and model measurement distributions: $g = H f + \eta, \quad \eta \sim q_n(\eta),$6 A tractable variational bound is obtained by introducing $g = H f + \eta, \quad \eta \sim q_n(\eta),$7 via $g = H f + \eta, \quad \eta \sim q_n(\eta),$8, yielding a multi-sample importance-weighted evidence lower bound (IWAE): $g = H f + \eta, \quad \eta \sim q_n(\eta),$9 Both $H \in \mathbb{R}^{m \times n}$0 (the main flow) and $H \in \mathbb{R}^{m \times n}$1 (the posterior flow) are parameterized as invertible architectures using coupling layers and $H \in \mathbb{R}^{m \times n}$2 invertible convolutions following RealNVP/Glow, enabling rapid density computation over both latent and data spaces (Kelkar et al., 2023).

The Riemannian AmbientFlow extends this architecture: the generator is a smooth invertible decoder $H \in \mathbb{R}^{m \times n}$3 parameterizing a model manifold, with explicit geometric structure extracted from the learned mapping (Diepeveen et al., 26 Jan 2026).

3. Training Procedure and Algorithmic Details

Training proceeds via stochastic optimization of the variational lower bound, with additional terms for optional regularization. Typical pseudocode structure:

• Sample minibatch of measurements $H \in \mathbb{R}^{m \times n}$4.
• For each $H \in \mathbb{R}^{m \times n}$5, draw $H \in \mathbb{R}^{m \times n}$6 noise vectors $H \in \mathbb{R}^{m \times n}$7, and map to $H \in \mathbb{R}^{m \times n}$8.
• Compute
  • $H \in \mathbb{R}^{m \times n}$9 via change of variables through $\eta$0,
  • $\eta$1 via $\eta$2 and
  • $\eta$3.
• Optionally apply sparsity penalty under a transform $\eta$4, hard-thresholded to $\eta$5-sparse representations for rank-deficient $\eta$6.
• The final objective is averaged over $\eta$7 samples, combining log-likelihood, regularization, and sparsity penalties; optimization uses Adam.

For Riemannian AmbientFlow, the geometric regularization term $\eta$8 encourages recovery of low-dimensional nondegenerate manifolds. An optional negative log-likelihood term for a small clean reference set (weight $\eta$9) can be added (Diepeveen et al., 26 Jan 2026).

4. Theoretical Guarantees

AmbientFlow inherits strong theoretical properties from flow-based generative models. Riemannian AmbientFlow establishes explicit error bounds under appropriate measurement and geometric assumptions:

• Under a restricted isometry property (RIP) condition on $p_\theta(f)$0, minimizers of the regularized variational loss achieve an Earth Mover's distance to the true data law bounded as

$p_\theta(f)$1

where $p_\theta(f)$2 is the variational tolerance and $p_\theta(f)$3 the RIP constant (Diepeveen et al., 26 Jan 2026).

• The learned decoder can be shown, under regularity constraints, to be bi-Lipschitz with explicitly controlled constants, enabling well-conditioned mappings suitable for inverse problems (Diepeveen et al., 26 Jan 2026).

The framework allows joint recovery of both a probabilistic normalizing-flow prior and a smooth manifold parameterization directly from corrupted (and optionally a handful of clean) samples, with the geometric structure—distances, geodesics, volume—readily computable via the pullback metric.

5. Numerical Experiments and Empirical Performance

AmbientFlow demonstrates robust performance across diverse domains:

• Synthetic 2D Gaussian mixtures: recovers multimodal structure from heavily noisy measurements;
• MNIST (blurred/noisy): generates denoised, sharp samples with Fréchet Inception Distance (FID) close to clean-trained flows;
• CelebA-HQ (faces): achieves FID $p_\theta(f)$4 40 vs. classical methods (BM3D, Wiener FID $p_\theta(f)$5 80); clean-trained flow yields FID $p_\theta(f)$6 20 (Kelkar et al., 2023);
• Stylized MRI: FID $p_\theta(f)$7 75 (AmbientFlow) vs. $p_\theta(f)$8 100 for classical PLS-TV reconstructions; qualitative radiomics feature matching superior to TV.

In downstream image reconstruction, AmbientFlow priors (learned solely from measurements) deliver MAP and MMSE reconstructions with RMSE/SSIM indistinguishable from flows trained on clean data, consistently outperforming traditional compressed sensing and TV regularization (Kelkar et al., 2023).

Riemannian AmbientFlow is validated on:

• Synthetic curved manifolds (sinusoidal in $p_\theta(f)$9), recovering accurate generative geometry;
• Noisy/blurry MNIST (14$q_f(f)$014): learned prior generates crisp digits, geodesic interpolation yields semantically smooth trajectories (Diepeveen et al., 26 Jan 2026).

6. Manifold Geometry and Applications in Inverse Problems

A central feature of Riemannian AmbientFlow is the extraction of an explicit manifold geometry from the generative model. Equip latent space $q_f(f)$1 with the pullback of Euclidean metric via the decoder $q_f(f)$2, leading to well-defined geodesics, Riemannian distances, and explicit volume forms. After training, an empirical “Riemannian AE” is constructed via tangent-space PCA at a barycenter, yielding a smooth reducer $q_f(f)$3 usable as a generative prior.

In inverse problems (e.g., reconstructing $q_f(f)$4 from $q_f(f)$5), this prior enables data-fidelity optimization in the latent space: $q_f(f)$6, outperforming TV baselines in both MSE and qualitative sharpness (Diepeveen et al., 26 Jan 2026).

7. Distinction from Other Notions of Ambient Flows

“Ill-posed” or “ambient flows” occasionally appear in other contexts (e.g., FlowMo-WM’s ambient drift variables (Jiang et al., 11 Jun 2026)). However, in the context of generative modeling and inverse problems, AmbientFlow and Riemannian AmbientFlow specifically refer to flow-based variational learning from corrupted measurement data, with rigorous probabilistic and geometric guarantees.

Summary Table: AmbientFlow and Its Riemannian Extension

Aspect	AmbientFlow	Riemannian AmbientFlow
Data type	Noisy/incomplete measurements	Noisy/incomplete, with manifold
Model architecture	Invertible flows	Invertible flows + manifold decoder
Key loss	Measurement marginal VLB/IWAE	VLB + geometric (Jacobian) penalty
Geometry extraction	Not explicit	Pullback Riemannian metric
Inverse problems	Bayesian inference via learned prior	Manifold-constrained optimization

AmbientFlow provides a principled route to unsupervised prior learning in domains where clean object data is unobtainable, and its Riemannian extension enables data-driven recovery of nonlinear geometry alongside generative modeling, with application in robust imaging, scientific data analysis, and beyond (Kelkar et al., 2023, Diepeveen et al., 26 Jan 2026).