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Adapting Noise to Data: Generative Flows from 1D Processes

Jannis Chemseddine, Gregor Kornhardt, Richard Duong, Gabriele Steidl

TL;DR

<3-5 sentence high-level summary> The paper addresses limitations of fixed Gaussian latent noise in flow-based generative models by enabling data-adaptive latent noise through learnable 1D processes. It develops a general framework that decomposes multivariate flows into independent 1D noises described by quantile functions, and shows three 1D processes (Wiener, Kac, and a uniform-MMD gradient flow) that can be learned jointly with Flow Matching or consistency models. The key idea is to represent 1D noise via quantile processes and jointly optimize the quantile maps and velocity field, resulting in shorter transport paths and better tail/support handling. Experiments on synthetic distributions and image datasets (MNIST, CIFAR-10) demonstrate the method's flexibility and effectiveness, including improved handling of heavy tails and complex supports. The work provides a new design space for generative modeling where learnable latent noise adapts to data and integrates with existing FM/IMM objectives.

Abstract

We introduce a general framework for constructing generative models using one-dimensional noising processes. Beyond diffusion processes, we outline examples that demonstrate the flexibility of our approach. Motivated by this, we propose a novel framework in which the 1D processes themselves are learnable, achieved by parameterizing the noise distribution through quantile functions that adapt to the data. Our construction integrates seamlessly with standard objectives, including Flow Matching and consistency models. Learning quantile-based noise naturally captures heavy tails and compact supports when present. Numerical experiments highlight both the flexibility and the effectiveness of our method.

Adapting Noise to Data: Generative Flows from 1D Processes

TL;DR

<3-5 sentence high-level summary> The paper addresses limitations of fixed Gaussian latent noise in flow-based generative models by enabling data-adaptive latent noise through learnable 1D processes. It develops a general framework that decomposes multivariate flows into independent 1D noises described by quantile functions, and shows three 1D processes (Wiener, Kac, and a uniform-MMD gradient flow) that can be learned jointly with Flow Matching or consistency models. The key idea is to represent 1D noise via quantile processes and jointly optimize the quantile maps and velocity field, resulting in shorter transport paths and better tail/support handling. Experiments on synthetic distributions and image datasets (MNIST, CIFAR-10) demonstrate the method's flexibility and effectiveness, including improved handling of heavy tails and complex supports. The work provides a new design space for generative modeling where learnable latent noise adapts to data and integrates with existing FM/IMM objectives.

Abstract

We introduce a general framework for constructing generative models using one-dimensional noising processes. Beyond diffusion processes, we outline examples that demonstrate the flexibility of our approach. Motivated by this, we propose a novel framework in which the 1D processes themselves are learnable, achieved by parameterizing the noise distribution through quantile functions that adapt to the data. Our construction integrates seamlessly with standard objectives, including Flow Matching and consistency models. Learning quantile-based noise naturally captures heavy tails and compact supports when present. Numerical experiments highlight both the flexibility and the effectiveness of our method.
Paper Structure (55 sections, 10 theorems, 113 equations, 16 figures, 1 algorithm)

This paper contains 55 sections, 10 theorems, 113 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline of the paper
  4. Flow Matching and Stochastic Processes
  5. Absolutely Continuous Curves in Wasserstein Space
  6. Flow Matching
  7. Stochastic Processes and Velocity Fields
  8. From One-Dimensional to Multi-Dimensional Flows
  9. Wiener Process and Diffusion Equation
  10. Kac Process and Damped Wave Equation
  11. Uniform Process and Gradient Flow of MMD Functional
  12. Scaled Latent Distributions
  13. Adapting Noise to Data
  14. Quantile Processes and Interpolants
  15. Quantile Processes.
  16. ...and 40 more sections

Key Result

Proposition 3

Let $\mu_t$ be given by comp, where the $\mu^i_t$ are absolutely continuous curves in $\mathbb{R}$ with velocity fields $v^i_t$. Then $\mu_t$ satisfies a multi-dimensional continuity equation eq:ce with a velocity field which decomposes into the univariate velocities

Figures (16)

  • Figure 1: FM via optimal coupling with Gaussian noise (left) and our learned noise (right). Samples from the latent distribution are black and from generated ones are red. The left image shows their transportation paths in green. Starting from the learned latent drastically shortened the paths. (Zoom in for the paths in the bottom image).
  • Figure 2: Three realisations of a standard Wiener process (left), the Kac process (middle), and the Uniform process (right), simulated until time $t=1$.
  • Figure 3: A generated trajectory from a Flow Matching model trained using the conditional density and velocity given by the Wiener process.
  • Figure 4: Samples (1M) from the distribution of the two-dimensional Kac process $\mathbf{K}_1$ for $(a,c)=(9,3)$. We can clearly observe the atomic component of its distribution derived in \ref{['kac-density-sing']}.
  • Figure 5: A generated trajectory from a Flow Matching model trained using the conditional density and velocity given by the Kac process with $(a,c)=(9,3)$.
  • ...and 11 more figures

Theorems & Definitions (21)

  • Remark 1: Relation to FM and diffusion
  • Remark 2: Optimal Coupling
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Example 7
  • Remark 8
  • Proposition 9
  • Theorem 10
  • ...and 11 more