Score-Jacobian spectral-gap analysis

methods / Theoretical / Analytical / Geometric analysis / Score-Jacobian spectral-gap analysis

Techniqueadvanced

Computes the eigenvalue (or singular-value) spectrum of the Jacobian of a diffusion model's learned score function at points along the denoising trajectory, using discontinuities (gaps) in that spectrum -- predicted by random-matrix/statistical-physics theory -- to reveal the presence, onset, and dimensionality of underlying data sub-manifolds as noise decreases.

Used in (3 observations)

Real trained DDPMs lose spectral-gap dimension smoothly as training-set size shrinks into a memorization regimemeasured

structure: Curvature profile of the representation manifold, Dimensional collapse · models: DDPM (MNIST, custom-trained, memorization-transition study), DDPM (Fashion-MNIST, custom-trained, memorization-transition study), DDPM (CIFAR-10, custom-trained, memorization-transition study), DDPM (CelebA-HQ, custom-trained, memorization-transition study), DDPM (LSUN-Church, custom-trained, memorization-transition study) · paper: Losing Dimensions: Geometric Memorization in Generative Diffusion

Diffusion-model score-Jacobian curvature maps onto a dense associative-memory energy landscape whose memorized/spurious/generalized transition scales with model capacitymeasured

structure: Curvature profile of the representation manifold, Dimensional collapse · models: DDPM (MNIST, custom-trained, memorization-transition study), DDPM (Fashion-MNIST, custom-trained, memorization-transition study), DDPM (CIFAR-10, custom-trained, memorization-transition study), DDPM (LSUN-Church, custom-trained, memorization-transition study) · paper: Memorization to Generalization: Emergence of Diffusion Models from Associative Memory

Real diffusion models trained on MNIST, CIFAR-10, and CelebA show score-Jacobian spectral gaps matching random-matrix-theory predictions for sub-manifold dimensionalitymeasured

structure: Curvature profile of the representation manifold · models: Score-based diffusion model (custom-trained, MNIST), Score-based diffusion model (custom-trained, CIFAR-10), Score-based diffusion model (custom-trained, CelebA) · paper: Manifolds, Random Matrices and Spectral Gaps: The Geometric Phases of Generative Diffusion