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Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold

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

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Introduces time-dependent log-density ridge manifolds from the smoothed empirical distribution and shows diffusion sampling follows a reach-align-slide mechanism relative to them, linked to directional training errors via random feature networks trained by GD.

Problem

Understanding where non-memorizing diffusion model samples lie relative to the data-dependent geometry of the training set, rather than population-level distributional closeness.

Approach

Defines a time-indexed family of log-density ridge manifolds from the smoothed empirical distribution and proves that reverse-time inference proceeds in three stages (reach the tube, align normally with error controlled by normal training loss, slide tangentially controlled by tangential training loss); RFNN+GD makes architecture vs. optimization effects explicit.

Key results

• On MNIST latent diffusion, mean ridge distance over 200 trajectories decreases through most of inference and stabilizes at a small floor after about 900 steps.
• On the two-point 2D dataset, normal error stays extremely small across weight schedules while tangential spread depends on the weight w(t), with w(t)=h_t^2 giving the largest spread.
• Tangential motion toward training data is slower than normal alignment and becomes negligible only near the end of inference on MNIST.

Surprising findings

• Tangential training loss alone does not determine final spread: w(t)=h_t^2 can have small tangential loss yet large tangential spread due to end-stage coefficient amplification.

Limitations

• Time discretization errors are not analyzed.
• Explicit training-level decomposition is only developed for random feature networks, not modern architectures.
• Several results rely on asymptotic order conditions as t approaches delta and T.

Experimental setup

Datasets: MNIST

Benchmarks: MNIST (digits 4 and 8, latent space)

Metrics: mean distance to ridge tangential error fraction of samples within radius 0.5 of target modes

Builds on: denoising score matching density ridges random feature models

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