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

Generative models typically rely on explicitly tracking the noise level to guide the sampling process. However, recent autonomous models successfully generate data using a single, time-invariant vector field that operates without explicit noise conditioning. This raises a fundamental paradox: what landscape are these networks actually optimizing when the noise level is treated as a random variable, and how can a bounded model remain stable near the data manifold where gradients typically diverge? We resolve this paradox by showing that autonomous generation is not merely “blind’’ denoising, but a specific form of Riemannian gradient flow on the Marginal Energy landscape, defined by integrating out the unknown noise levels. Through a novel relative energy decomposition, we demonstrate that while the raw Marginal Energy contains a severe geometric singularity normal to the data manifold, the learned field implicitly incorporates a local conformal metric. This metric perfectly preconditions the singularity, turning an infinitely deep potential well into a stable attractor. Finally, we establish strict structural stability conditions for autonomous sampling. We prove that standard noise-prediction parameterizations structurally fail due to an inherent high gain amplification of the estimation errors. In contrast, velocity-based models are inherently stable.