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A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields

Giacomo Greco

TL;DR

The paper develops an infinite-dimensional extension of score-based diffusion models by embedding the forward noising mechanism in a Gamma-calculus/Dirichlet-form framework on a Hilbert space, using the Cameron–Martin space to define the noise and Malliavin calculus to characterize the score. A central novelty is the abstract time-reversal result that identifies the Gamma-score with a conditional expectation of the initial state, enabling a backward diffusion procedure in Hilbert spaces under a finite entropy assumption. The authors derive entropic convergence bounds in KL divergence that are governed by the Cameron–Martin norm and demonstrate the approach on spherical random fields with Whittle–Matérn noise, including numerical simulation details. This provides a principled, dimension-free foundation for infinite-dimensional SGMs and opens avenues for non-Gaussian noise targets and manifold-valued data modeling with theoretical guarantees.

Abstract

We adopt a Gamma and Malliavin Calculi point of view in order to generalize Score-based diffusion Generative Models (SGMs) to an infinite-dimensional abstract Hilbertian setting. Particularly, we define the forward noising process using Dirichlet forms associated to the Cameron-Martin space of Gaussian measures and Wiener chaoses; whereas by relying on an abstract time-reversal formula, we show that the score function is a Malliavin derivative and it corresponds to a conditional expectation. This allows us to generalize SGMs to the infinite-dimensional setting. Moreover, we extend existing finite-dimensional entropic convergence bounds to this Hilbertian setting by highlighting the role played by the Cameron-Martin norm in the Fisher information of the data distribution. Lastly, we specify our discussion for spherical random fields, considering as source of noise a Whittle-Matérn random spherical field.

A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields

TL;DR

The paper develops an infinite-dimensional extension of score-based diffusion models by embedding the forward noising mechanism in a Gamma-calculus/Dirichlet-form framework on a Hilbert space, using the Cameron–Martin space to define the noise and Malliavin calculus to characterize the score. A central novelty is the abstract time-reversal result that identifies the Gamma-score with a conditional expectation of the initial state, enabling a backward diffusion procedure in Hilbert spaces under a finite entropy assumption. The authors derive entropic convergence bounds in KL divergence that are governed by the Cameron–Martin norm and demonstrate the approach on spherical random fields with Whittle–Matérn noise, including numerical simulation details. This provides a principled, dimension-free foundation for infinite-dimensional SGMs and opens avenues for non-Gaussian noise targets and manifold-valued data modeling with theoretical guarantees.

Abstract

We adopt a Gamma and Malliavin Calculi point of view in order to generalize Score-based diffusion Generative Models (SGMs) to an infinite-dimensional abstract Hilbertian setting. Particularly, we define the forward noising process using Dirichlet forms associated to the Cameron-Martin space of Gaussian measures and Wiener chaoses; whereas by relying on an abstract time-reversal formula, we show that the score function is a Malliavin derivative and it corresponds to a conditional expectation. This allows us to generalize SGMs to the infinite-dimensional setting. Moreover, we extend existing finite-dimensional entropic convergence bounds to this Hilbertian setting by highlighting the role played by the Cameron-Martin norm in the Fisher information of the data distribution. Lastly, we specify our discussion for spherical random fields, considering as source of noise a Whittle-Matérn random spherical field.
Paper Structure (10 sections, 5 theorems, 100 equations, 4 figures, 2 algorithms)

This paper contains 10 sections, 5 theorems, 100 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The point of view of $\Gamma$-calculus and Dirichlet forms
  3. Main contributions
  4. Comparison with existing literature.
  5. Noising dynamics in Hilbertian setting
  6. Denoising via abstract time-reversal
  7. $\Gamma$-Score approximation and sampling scheme
  8. Generative model for spherical random fields
  9. Numerical simulation
  10. Convergence results for finite-dimensional approximations

Key Result

Theorem 1

Assume $\mu_{\mathrm{data}}\ll\mathfrak{m}$. For any $u\in\mathcal{PC}$ and for any $t>0$ we have

Figures (4)

  • Figure 1: Simulation of the forward noising process at different time-steps.
  • Figure 2: Plot of empirical and theoretical power spectra
  • Figure 3: Plot of averaged (over last 30 steps), empirical and theoretical power spectra
  • Figure 4: Ratio between empirical $\hat{C}_\ell$ and theoretical $C_\ell$ (on the left) and ratio between averaged empirical $\hat{C}_\ell$ and theoretical $C_\ell$ (on the right)

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 1: $\Gamma$-score identification
  • proof
  • Remark 3
  • Lemma 1: Lemma 1.4.2 in Nualart2006
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • ...and 3 more