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

• URL: http://arxiv.org/abs/2602.18428v1
• Date: Fri, 20 Feb 2026 18:49:00 GMT
• Title: The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning
• Authors: Mojtaba Sahraee-Ardakan, Mauricio Delbracio, Peyman Milanfar,
• Abstract summary: We study autonomous (noise-agnostic) generative models, such as Equilibrium Matching and blind diffusion.<n>We prove that generation using autonomous models is not merely blind denoising.<n>We also establish the structural stability conditions for sampling with autonomous models.
• Score: 20.547812775989808
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Autonomous (noise-agnostic) generative models, such as Equilibrium Matching and blind diffusion, challenge the standard paradigm by learning a single, time-invariant vector field that operates without explicit noise-level conditioning. While recent work suggests that high-dimensional concentration allows these models to implicitly estimate noise levels from corrupted observations, a fundamental paradox remains: what is the underlying landscape being optimized when the noise level is treated as a random variable, and how can a bounded, noise-agnostic network remain stable near the data manifold where gradients typically diverge? We resolve this paradox by formalizing Marginal Energy, $E_{\text{marg}}(\mathbf{u}) = -\log p(\mathbf{u})$, where $p(\mathbf{u}) = \int p(\mathbf{u}|t)p(t)dt$ is the marginal density of the noisy data integrated over a prior distribution of unknown noise levels. We prove that generation using autonomous models is not merely blind denoising, but a specific form of Riemannian gradient flow on this Marginal Energy. Through a novel relative energy decomposition, we demonstrate that while the raw Marginal Energy landscape possesses a $1/t^p$ singularity normal to the data manifold, the learned time-invariant field implicitly incorporates a local conformal metric that perfectly counteracts the geometric singularity, converting an infinitely deep potential well into a stable attractor. We also establish the structural stability conditions for sampling with autonomous models. We identify a ``Jensen Gap'' in noise-prediction parameterizations that acts as a high-gain amplifier for estimation errors, explaining the catastrophic failure observed in deterministic blind models. Conversely, we prove that velocity-based parameterizations are inherently stable because they satisfy a bounded-gain condition that absorbs posterior uncertainty into a smooth geometric drift.

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