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Spectral Diffusion Models on the Sphere

Pierpaolo Brutti, Claudio Durastanti, Francesco Mari

TL;DR

This work extends score-based diffusion modeling to spherical data by formulating forward and reverse SDEs directly in the spherical harmonic spectral domain. It shows that the spherical DFT maps spatial Brownian motion to a constrained, non-isotropic Gaussian, necessitating a spherical mirrored Brownian noise model and a covariance $\Sigma$ in the spectral diffusion dynamics. The authors derive a matrix-based spherical DFT framework with $U = Y^{\mathsf H}Q$, establish the forward/reverse SDEs for spectral coefficients, and analyze the relationship between spatial and spectral score-matching losses, proving they are not equivalent due to geometry-induced bias. Numerical illustrations on band-limited spheres, including MNIST-Sphere, demonstrate numerical viability and reveal the geometry-driven differences in diffusion behavior while yielding comparable sample quality to spatial-domain models. This framework enables principled generative modeling of spherical data with potential applications in cosmology, earth science, and computer graphics.

Abstract

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. Extending spectral diffusion approaches to spherical data, however, raises nontrivial geometric and stochastic issues that are absent in the Euclidean setting. In this work, we develop a diffusion modeling framework defined directly on finite-dimensional spherical harmonic representations of real-valued functions on the sphere. We show that the spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with deterministic, generally non-isotropic covariance. This induces modified forward and reverse-time stochastic differential equations in the spectral domain. As a consequence, spatial and spectral score matching objectives are no longer equivalent, even in the band-limited setting, and the frequency-domain formulation introduces a geometry-dependent inductive bias. We derive the corresponding diffusion equations and characterize the induced noise covariance.

Spectral Diffusion Models on the Sphere

TL;DR

This work extends score-based diffusion modeling to spherical data by formulating forward and reverse SDEs directly in the spherical harmonic spectral domain. It shows that the spherical DFT maps spatial Brownian motion to a constrained, non-isotropic Gaussian, necessitating a spherical mirrored Brownian noise model and a covariance in the spectral diffusion dynamics. The authors derive a matrix-based spherical DFT framework with , establish the forward/reverse SDEs for spectral coefficients, and analyze the relationship between spatial and spectral score-matching losses, proving they are not equivalent due to geometry-induced bias. Numerical illustrations on band-limited spheres, including MNIST-Sphere, demonstrate numerical viability and reveal the geometry-driven differences in diffusion behavior while yielding comparable sample quality to spatial-domain models. This framework enables principled generative modeling of spherical data with potential applications in cosmology, earth science, and computer graphics.

Abstract

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. Extending spectral diffusion approaches to spherical data, however, raises nontrivial geometric and stochastic issues that are absent in the Euclidean setting. In this work, we develop a diffusion modeling framework defined directly on finite-dimensional spherical harmonic representations of real-valued functions on the sphere. We show that the spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with deterministic, generally non-isotropic covariance. This induces modified forward and reverse-time stochastic differential equations in the spectral domain. As a consequence, spatial and spectral score matching objectives are no longer equivalent, even in the band-limited setting, and the frequency-domain formulation introduces a geometry-dependent inductive bias. We derive the corresponding diffusion equations and characterize the induced noise covariance.
Paper Structure (20 sections, 6 theorems, 85 equations, 1 figure, 1 table)

This paper contains 20 sections, 6 theorems, 85 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Score-Based Generative Modeling in Euclidean Space
  4. Forward Diffusion.
  5. Reverse Diffusion.
  6. Score Estimation.
  7. Harmonic Analysis on the Sphere
  8. Spherical Harmonics.
  9. Band-Limited Functions and Sampling.
  10. Probability Densities in the Spectral Domain.
  11. Diffusion Models in the Frequency Domain for Spherical Data
  12. Matrix expression for the spherical DFT
  13. Diffusion SDEs
  14. Relationship between Score Matching Losses
  15. Numerical Illustrations
  16. ...and 5 more sections

Key Result

Lemma 3.3

Let $f \in L^2(\mathbb{S}^2)$ be $L$-band-limited, and let be its samples on the equiangular grid. Define the vector of spatial samples and let $\hat{\mathbf{a}} \in \mathbb{C}^{L^2}$ be the vector of spherical Fourier coefficients ordered as Then the discrete spherical Fourier transform can be expressed as the matrix--vector product where $Y$ is the spherical harmonics matrix and $Q$ the samp

Figures (1)

  • Figure 1: Comparison between the empirical covariance matrix of $\varphi[{\mathbf{\tilde{v}}}(t)]$, for $t=0$, estimated from $5\times10^{4}$ i.i.d samples (left), and the theoretical covariance matrix $\Sigma$ (right), with maximum bandlimit $L=4$.

Theorems & Definitions (20)

  • Remark 2.1
  • Definition 3.1: Spherical harmonics matrix
  • Definition 3.2: Sample weights matrix
  • Lemma 3.3: DFT of a real-valued spherical map
  • Lemma 3.4: Pseudoinverse of the spherical DFT operator $U$
  • Remark 3.5
  • Proposition 3.6: Isometry between the $L$-band limited spherical signal space and the Fourier coefficients space
  • Definition 3.7: Spherical mirrored Brownian motion
  • Lemma 3.8: Image of Brownian motion under the spherical DFT
  • Remark 3.9: Densities and scores for constrained spherical Fourier coefficients
  • ...and 10 more