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The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning

NeurIPS 2026 · Image generation › Fast diffusion sampling · approach: Diffusion theory and design space · 0 citations (OpenAlex)

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Proposes a marginal-energy framework explaining why noise-agnostic (autonomous) generative models can sample: their time-invariant field is a preconditioned Riemannian gradient flow on the marginal energy, with velocity parameterizations stable and noise prediction unstable.

Problem

Autonomous generative models learn a single time-invariant vector field without noise-level conditioning; it is unclear what energy landscape they optimize and how a bounded network stays stable near the data where gradients of the marginal energy diverge.

Approach

Defines the marginal energy -log p(u) integrated over a prior on noise levels, shows the autonomous field decomposes into a natural gradient term, a covariance transport correction and linear drift, with an effective gain acting as a conformal preconditioner; argues noise-prediction parameterizations have a high-gain "Jensen Gap" while velocity parameterizations have bounded gain.

Key results

• On CIFAR-10 (numbers taken from prior work), DDIM noise-prediction without t gets FID 40.90 versus Flow Matching 2.61 and uEDM 2.23 under the same blind setting.
• Qualitatively, DDPM Blind fails with high-frequency artifacts on CIFAR-10, SVHN and Fashion MNIST while Flow Matching Blind matches conditional counterparts.
• In a 2D concentric-circles toy, autonomous Flow Matching yields clean rings by D=8 while DDPM Blind is noisier; both produce clean samples at D=128.

Surprising findings

• Even the structurally unstable DDPM Blind eventually produces clean samples in very high ambient dimension (D=128), since estimation error vanishes faster than the gain diverges.

Limitations

• Stability analysis gives sufficient conditions for instability and only necessary conditions for stability; not strictly sufficient.
• Empirical verification is small-scale (10,000 training steps, toy U-Net) and CIFAR-10 numbers are borrowed from prior work rather than reproduced.
• Several proofs are heuristic (Laplace approximation, asymptotic approximations of the field near the manifold, and assumptions such as uniform or flat priors on time).

Experimental setup

Datasets: CIFAR-10 SVHN Fashion MNIST 2D concentric circles (synthetic)

Benchmarks: CIFAR-10

Metrics: FID qualitative sample inspection

Baselines: DDPM Blind DDPM Conditional Flow Matching Conditional DDIM Flow Matching uEDM

Builds on: Equilibrium Matching Flow Matching DDPM Tweedie's formula Blind diffusion

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