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Inverse Flow and Consistency Models

Yuchen Zhang, Jian Zhou

TL;DR

Inverse Flow (IF) reframes inverse generation as learning a mapping from observed $\mathbf{x}_1$ to unobserved $\mathbf{x}_0$, enabling denoising without ground truth data. It introduces two methods, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM), with a simulation-free objective via Generalized Consistency Training and theoretical links to continuous-time generative models (ODE/SDE) such as diffusion, CFMs, and CM. Empirically, IFM/ICM outperform prior unsupervised denoising methods on synthetic, semi-synthetic, and real-world data, including fluorescence microscopy and single-cell genomics, while handling flexible noise distributions beyond linear corruption. The work broadens the applicability of continuous-time generative modeling to inversion problems, offering practical denoising capabilities without clean training data and with improved computational efficiency through ICM.

Abstract

Inverse generation problems, such as denoising without ground truth observations, is a critical challenge in many scientific inquiries and real-world applications. While recent advances in generative models like diffusion models, conditional flow matching, and consistency models achieved impressive results by casting generation as denoising problems, they cannot be directly used for inverse generation without access to clean data. Here we introduce Inverse Flow (IF), a novel framework that enables using these generative models for inverse generation problems including denoising without ground truth. Inverse Flow can be flexibly applied to nearly any continuous noise distribution and allows complex dependencies. We propose two algorithms for learning Inverse Flows, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM). Notably, to derive the computationally efficient, simulation-free inverse consistency model objective, we generalized consistency training to any forward diffusion processes or conditional flows, which have applications beyond denoising. We demonstrate the effectiveness of IF on synthetic and real datasets, outperforming prior approaches while enabling noise distributions that previous methods cannot support. Finally, we showcase applications of our techniques to fluorescence microscopy and single-cell genomics data, highlighting IF's utility in scientific problems. Overall, this work expands the applications of powerful generative models to inversion generation problems.

Inverse Flow and Consistency Models

TL;DR

Inverse Flow (IF) reframes inverse generation as learning a mapping from observed to unobserved , enabling denoising without ground truth data. It introduces two methods, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM), with a simulation-free objective via Generalized Consistency Training and theoretical links to continuous-time generative models (ODE/SDE) such as diffusion, CFMs, and CM. Empirically, IFM/ICM outperform prior unsupervised denoising methods on synthetic, semi-synthetic, and real-world data, including fluorescence microscopy and single-cell genomics, while handling flexible noise distributions beyond linear corruption. The work broadens the applicability of continuous-time generative modeling to inversion problems, offering practical denoising capabilities without clean training data and with improved computational efficiency through ICM.

Abstract

Inverse generation problems, such as denoising without ground truth observations, is a critical challenge in many scientific inquiries and real-world applications. While recent advances in generative models like diffusion models, conditional flow matching, and consistency models achieved impressive results by casting generation as denoising problems, they cannot be directly used for inverse generation without access to clean data. Here we introduce Inverse Flow (IF), a novel framework that enables using these generative models for inverse generation problems including denoising without ground truth. Inverse Flow can be flexibly applied to nearly any continuous noise distribution and allows complex dependencies. We propose two algorithms for learning Inverse Flows, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM). Notably, to derive the computationally efficient, simulation-free inverse consistency model objective, we generalized consistency training to any forward diffusion processes or conditional flows, which have applications beyond denoising. We demonstrate the effectiveness of IF on synthetic and real datasets, outperforming prior approaches while enabling noise distributions that previous methods cannot support. Finally, we showcase applications of our techniques to fluorescence microscopy and single-cell genomics data, highlighting IF's utility in scientific problems. Overall, this work expands the applications of powerful generative models to inversion generation problems.

Paper Structure

This paper contains 41 sections, 4 theorems, 61 equations, 11 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Continuous-time generative models
  4. Conditional flow matching
  5. Consistency models
  6. Diffusion models
  7. Denoising applications of diffusion models
  8. Ambient diffusion and GSURE-diffusion
  9. Denoising without ground truth
  10. Inverse Flow and Consistency Models
  11. Inverse Flow Matching
  12. Simulation-free Inverse Flow with Inverse Consistency Model
  13. Generalized Consistency Training
  14. Inverse Consistency Models
  15. Experiments
  16. ...and 26 more sections

Key Result

Theorem 1

Assume that the noise distribution $p(\mathbf{x}_{1}\mid \mathbf{x}_{0})$ satisfies the condition that, for any noisy data distribution $p(\mathbf{x}_{1})$ there exists only one probability distribution $p(\mathbf{x}_{0})$ that satisfies $p(\mathbf{x}_{1})=\int p(\mathbf{x}_{1}\mid \mathbf{x}_{0})p(

Figures (11)

  • Figure 1: Inverse flow enables adapting the family continuous-time generative models for solving inverse generation problems. For inverse flow matching and inverse consistency model, $\mathbf{x}_0$ indicates unobserved data and $\mathbf{x}_1$ indicates observed data. For conditional flow matching and consistency model, $\mathbf{x}_0$ indicates data and $\mathbf{x}_1$ indicates variable from the prior distribution. Inverse flow algorithms modify continuous-time generative models to solve the inverse generation problem of recovering unobserved $\mathbf{x}_0$ from $\mathbf{x}_1$ by replacing the unobserved $p(\mathbf{x}_0)$ with generated $q(\mathbf{x}_0)$ within the training loop.
  • Figure 2: Demonstration of inverse flow algorithms on synthetic datasets. Top panel shows an application to inverting Navier-Stokes fluid dynamics simulation color indicating the difference between the input state and the initial state. Bottom panel shows a denoising application on 8-gaussians dataset with input (black) and denoised data (blue) connected with lines.
  • Figure 3: Denoising results for fluorescence microscopy images with PSNR labelled.
  • Figure 4: Denoising single-cell RNA-seq data with ICM improves resolution for cell types and developmental trajectories. The top two principal components are visualized. Top panel: results for zeisel_molecular_2018. Bottom panel: results for hochgerner_conserved_2018, Astro: astrocytes, RGL: radial glial cells, IPC: intermediate progenitor cells, OPC: oligodendrocyte precursor cells, MOL: mature oligodendrocytes; NFOL: newly formed oligodendrocytes, GABA: GABAergic neurons, GC: granule cells, Pyr: pyramidal neurons.
  • Figure 5: Original Prediction of inverting Navier-Stokes fluid dynamics simulation, color indicating horizontal velocity.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2